Hi You might find this paper interesting and controversial . It proves

http://gamahucherpress.yellowgum.com/wp-content/uploads/All-things-are-possible.pdf


1)Mathematics/science end in contradiction. When mathematics/science end in contradiction it is proven in logic that you can prove anything you want in mathematics ie you can prove Fermat's last theorem and you can disprove Fermat's last theorem

2) The paper also proves in logic that all opinions are equally valid ie feminism is valid and anti-feminism is valid. Also the opinions for gay marriage are valid: opinions for gay marriage and against gay marriage are equally valid. Logic shows: the opinions of the left are as equally valid as the opinions of the center and right

Sara242- see Godel's Incompleteness Theorem

http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems

I believe this is the theorem that Colin Leslie Dean uses to indicate that Maths is inconsistent.

The incompleteness theorem is sometimes thought to have severe consequences for the program of logicism proposed by Gottlob Frege and Bertrand Russell, which aimed to define the natural numbers in terms of logic (Hellman 1981, p. 451–468). Bob Hale and Crispin Wright argue that it is not a problem for logicism because the incompleteness theorems apply equally to first order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem.

The Wiki article refers to a comparable problem "the liars paradox" which is also inconsistent.

Science tends to use Maths as a tool to extrapolate upon empirical sources not as an a priori source.

Using Science for invalid purposes is known as Scientism. But I like your ideas Sara242.

It may be controversial among Arts students, but mathematicians would be united in declaring it to be total crap. Mathematics is totally consistent. The argument otherwise proves nothing because it relies on the false premise that a number with infinite recursion is not a finite number.

"The argument otherwise proves nothing because it relies on the false premise that a number with infinite recursion is not a finite number."

0.999... is non finite it cannot be counted-it has no end

another way

fact is 1 is an integer 0.999.. is a non-integer
when an integer(1) = a non-integer(0.999...)
we have a contradiction in maths

You're talking about countable (integers, rationals) and uncountable (real) numbers. I think this was one of Hilbert's problems- the second one from memory. This problem was proved by Georg Cantor who I believe killed himself by cyanide though I could be wrong.

This demonstrates that even experts can struggle with these ideas. In Maths it's easy to make incorrect conclusions to results especially for the uninitiated- this doesn't mean that the uninitiated cannot make valuable contributions- but perhaps some care is required.

1 is a finite number it stops
A finite decimal is one that stops, like 0.157
A non-finite decimal like 0.999... does not stop
when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction

another way
1 is an integer a whole number
0.888... is a non-integer it is not a whole number
0.999... is a non-integer not a whole number
when a integer 1 =a non-integer 0.999... maths ends in contradiction

This is the concept of convergent series/ sequences I guess.
I guess it can seem like a contradiction- even mathematicians don't know everything about maths- I don't have a good answer for you. I suggest you read more widely. If you can prove something you could win a substantial maths prize. Good luck Sara242.

Sara242 - Are you sure your name isn't Toni Lavis? :)

>0.999... is non finite it cannot be counted-it has no end
Wrong! Being non countable does not something from being finite.

An infinite series producing a finite sum is quite common in mathematics. I suggest you study it, or maybe read a mathematics textbook, rather than making ridiculous claims about a subject you clearly know very little about.

>another way
>fact is 1 is an integer 0.999.. is a non-integer
>when an integer(1) = a non-integer(0.999...)
>we have a contradiction in maths

That's not an actual contradiction; merely an apparent contradiction.
It's evidence that at least one of the following tow assumptions you have made is FALSE:
1) numbers with integer values cease to be integers when they're expressed in other ways
2) integers can't be equal to non integers.

I'm not sufficiently well versed in semantics to tell you with absolute certainty which one you have got wrong, but I think it's the first. AIUI integers are defined by how they can be expressed rather than who they are expressed.

It might help you understand if you expressed your decimals as fractions instead. You're effectively saying that integers are whole numbers not fractions, so 1 can't be an integer because it's equal to 9/9. You're contradicting yourself, but the mathematics remains consistent.

Aiden- I like the series 1+1/2+1/4... = 2

sara42: I'm guessing that you've never studied maths at uni level. If you had then you should be familiar with the various foundations commonly used for mathematics such as Zermelo-Fraenkel set theory ("ZFC"- when containing the axiom of choice) and also should have seen how to construct the real numbers within such a system.

If you study and understand the standard constructions of the real numbers then you will see that your "contradiction" doesn't make any sense. The main problem with it is, is that you are comparing integers and real numbers. But these are completely separately constructed things of maths. However you can embed the integers* into the real numbers in a 1-to-1 fashion. Ie: given any integer and its properties you can produce a unique real number that has corresponding properties. Specifically, when representing reals with infinite decimal expansions, for any integer you can associate a corresponding real which is the same written representation but with an additional infinite number of zeros after the decimal point. eg: the integer 1 maps to the real 1.00000..., 2 -> 2.0000...., -3 -> -3.0000..., also similar for any finite decimal fraction, eg: 0.5 -> 0.500000..., -0.11 -> -0.1100000, etc. See how ALL numbers now have infinite representations and not finite ones.

Also, it is a provable fact that the real number represented by: 0.99999999.. is the same as that represented by 1.00000... . The formal proof is quite advanced for those who've never studied university maths but simple enough that those of average intelligence can grasp it. If you really want to understand it would only take a few months of study. (If you've ever studied calculus at high school you are well on the way because one of the methods used to construct the real numbers does so with a process similar to limits used in calculus- called a "Cauchy sequence": https://en.wikipedia.org/wiki/Construction_of_the_real_numbers and https://en.wikipedia.org/wiki/Cauchy_sequence).

---
*: actually, typically you embed the integers into the rationals and then embed the rationals into the reals

Hmh, some educated but not self-employed by the gist of it, amusing themselves.

Individual- You're right of course. Idle hands.

Sara242-
There's the Boundedness Theorem that seems related here- though I'm not so good at "Pure Maths".

Try this link
http://socratic.org/precalculus/functions-defined-and-notation/boundedness

Or this one
http://math.stackexchange.com/questions/1365882/boundedness-theorem-for-continuous-functions-question

Some very good answers being posted though on this thread

Think about the base you are using. 0.11111... in base 9 is 0.1.

"Logic proves: All opinions -for and against are equally valid."

Illogical.

"Logic proves: All opinions -for and against are equally valid."

Logic would be that some opinions are valid & some not.

Aidan: You said in your first post, "Mathematics is totally consistent. ".
This is not a provable! (This is what is known as Godel's First Incompleteness Theorem.) So it may well be true, but we will never be able to prove it. In other words, you are stating it as a matter of faith not as a fact of maths.

Getting back to sara242's statement and where it actually fails to prove an inconsistency:
sara242 first "proof" states
1) fact is 1 is an integer 0.999.. is a non-integer
2) when an integer(1) = a non-integer(0.999...) we have a contradiction in maths"
- believe it or not both these statements are perfectly true statements about maths!
However, within standard maths it is NEVER the case that the integer(1) equals the non-integer (ie real) 0.9999 since you cannot compare integers to reals. There is no such equality operator- we only have equality operators that compare natural numbers to other naturals, integers to integers and so on- ie: you can only compare numbers of the same type. So even though (2) is a logically valid implication it is not logically sound (since the premise is never true) so we cannot draw the conclusion. That is we cannot state the conclusion by itself as a true statement.
(But note we can still state the overall (2) statement as true. Another example of this sort of thing is the statement: if squares are circles then 2+2=5. This statement is perfectly true, however squares as far as we know are not circles so we can't use this implication to claim anything about 2+2 equaling 5.)

Logic proves: All opinions -for and against are equally valid.

Firstly opinions are just opinions, separate from facts but sometimes some opinions are more factual than other opinions so assuming all opinions are equal in any way seems to me like a statement that's utterly false and invalid.

thinkabit,
> This is not a provable! (This is what is known as Godel's First Incompleteness Theorem.)
ITYM Godel's Second Incompleteness theorem, though that's not quite what it says.

>Getting back to sara242's statement and where it actually fails to prove an inconsistency:
> sara242 first "proof" states
> 1) fact is 1 is an integer 0.999.. is a non-integer
> 2) when an integer(1) = a non-integer(0.999...) we have a contradiction in maths"
> - believe it or not both these statements are perfectly true statements about maths!

No I don't believe it, and I'm wondering where you got such a ridiculous idea?

AIUI the mathematical definition of "integer" is much broader than the computing definition, and includes anything that is equal to an integer, so 0.999... is indeed an integer. And if I'm wrong on the semantics and the mathematical definition of an integer doesn't include anything that's equal to an integer, that would mean that an integer being equal to a non integer would not be a contradiction.

> However, within standard maths it is NEVER the case that the integer(1) equals the
> non-integer (ie real) 0.9999 since you cannot compare integers to reals.
ROFL
1=0.999...
I have just disproved your claim!

And more generally, integers are a subset of reals.

> There is no such equality operator- we only have equality operators that compare natural numbers to
> other naturals, integers to integers and so on- ie: you can only compare numbers of the same type.
No, that's not standard mathematics, that's some computer programming languages. But even those with that limitation generally have functions like parseInt to enable comparison.

Aidan: What you've written about integers and reals is VERY wrong!

An integer is a VERY different object to a real number and you cannot compare the two directly. When a mathematicians write something like 1=0.999 they are assuming the "1" is a real and not an integer. If you like a better way to write it is 1.000... = 0.9999... The integers are definitely NOT a subset of the reals. However, you can create a 1-to-1 mapping from the integers into the reals, eg: 1 -> 1.0000, 2->2.000.. etc. And under this mapping the arithmetic properties of the integers will be preserved on the reals, such as the integer addition of any two integers will correspond to the real addition of the reals to the integers map to, eg 1+1=2 under integer addition and 1.000..+1.000.. = 2.000 under real addition (this sort of thing is called a homomorphism : https://en.wikipedia.org/wiki/Homomorphism).

Let me explicitly demonstrate how different the natural number 1, the integer 1, the rational 1/1 and the real 1.000 are. I will give an outline of how to construct each number within the Zermelo-Fraenkel set theory, von Neumann hierarchy of sets and Cauchy sequence real number construction paradigm. >

--continued below --

<

Aidan: What you've written about integers and reals is VERY wrong!
thinkabit,
That's what you get when an Academic orientated attempts to grasp logic.

-continued from above-

-the natural number 1 is the set {{}} when we take the popular/standard approach and use the von Neumann hierarchy of sets (https://en.wikipedia.org/wiki/Von_Neumann_universe) as a basis for our number systems (ie, by definition 1 is the power set of the set for zero or equivalently it is set that contains the empty set since the natural number 0 is the empty set)

-the integer number 1 is the set that is the equivalence class of all ordered pairs of natural numbers <m,n> induced by the relation: <m,n> is related to <1,0> if and only if m+0 = n+1, where the addition operator + is the standard addition of the natural numbers.
So loosely the integer 1 is something like { <1,0>, <2,1>, <3,2>, <4,3>, ... } where each ordered pair of natural number <m,n> is the set {{m},{m,n}} and each natural number is that given by the von Neumann hierarchy above (ie:0={}, 1={{}}, 2={ {},{{}} }, 3={{},{{}},{{},{{}}}, etc..).
Putting this all together gives integer 1 = { {{{{}}},{{{}},{{}}}}, {{{{},{{}}}},{{{},{{}}},{{}}}}, {{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}},{{},{{}}}}}, ...} which clearly is DEFINITELY NOT equal to the set {{}} which is the natural number 1.
Note: observe that the integer number 1 is actually an infinite set!

-the rational number 1 is the set that is the equivalence class of all ordered pairs of integer numbers <m,n> induced by the relation: <m,n> is related to <1,1> if and only if m*1 = n*1, where the multiplication operator * is the standard integer multiplication of integer numbers.
So loosely the rational number 1 is something like {<1,1>,<-1,-1>,<2,2>,<-2,2>,<3,3>,<-3,3>...} when each ordered pair of integers <m,n> is the set {{m},{m,n}} and each integer is a set as explained above

--continued below-

--continued from above-

Putting this together gives, rational 1 is { {{{ {{{{}}},{{{}},{{}}}}, {{{{},{{}}}},{{{},{{}}},{{}}}}, {{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}},{{},{{}}}}}, ...}},{{ {{{{}}},{{{}},{{}}}}, {{{{},{{}}}},{{{},{{}}},{{}}}}, {{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}},{{},{{}}}}}, ...}.{ {{{{}}},{{{}},{{}}}}, {{{{},{{}}}},{{{},{{}}},{{}}}}, {{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}},{{},{{}}}}}, ...}}}, .... }. And the bit I wrote explicitly this is just the first element <1,1> ! But worse yet there are an infinite number of these monstrosities. ie: there are an infinity of infinity of sets in the definition of the rational number 1 and it is clearly not the same as the integer number 1.

- the real number 1.0000... is stupendously complicated set. A god-awful mess that is hard explain and simply impossible to even begin to explicitly write out by hand. Basically the real number 1.000.. is the collection of all Cauchy sequences of rationals that approach 1. So just how big is this set? Well there are an infinity of Cauchy sequences that approach 1 and each of these sequences has an infinite number of rationals in it and as explained above each rational is built upon an infinity of an infinity of sets. Ie, the real number 1.000.. is built upon an infinity of an infinity of an infinity of an infinity of sets and DEFINITELY NOT equal the set for the rational 1/1 nor the integer 1 nor the set {{}} which is the natural number 1.

"Pure Maths"- Yuck. I prefer Applied Maths.
But ThinkABit you made it interesting. Kudos.
I don't mind pure maths in small doses :)

Good comments from everyone here. Kudos.

thinkabit,
How do you get through the day with these things on your mind or are you a hard drive ? :-)

A real number is defined as a non-empty set, S, of rational numbers, such that if:

1) x and y are rational numbers¹; AND
2) x is in S; AND
3) y is less than² x,

then y is also in S.

http://en.wikipedia.org/wiki/Dedekind_cut

A rational number is converted into a real number by mapping it to the set of all smaller rational numbers, excluding itself.

Both the sets described by "1" (or "1/1") and "0.999999..." contain exactly the same members³, hence they are the same.

---
¹ i.e. they can each be expressed as a fraction of two integers, "a/b" and "c/d" respectively
² i.e. c×b < a×d
³ i.e. all the rational numbers "u/v" such that 'u' and 'v' are integers, v > 0 and u < v

Yuyutsu,
Trust you to throw in your 2xBob. :-)

I just re-read what I posted previously today and should correct something: Where is asked "So just how big is this set?" I should really be asking "So how complicated is this set?"

Yuyutsu:
The equivalence class of Cauchy sequences approaching the same limit is the standard construction method taught these days (atleast in Austrlia a few decades ago when I was studying this stuff). Dedekind cuts were certainly a popular construction over a hundred years ago. However I suspect that it fell out of favour as since students were studying more and more rigorous versions of calculus at an early stage where they were exposed to the technical concepts of detla-eplison limits and then when for those that moved on to studying the deeper concepts of the real number line, such as its construction, the professors could leverage this established knowledge base by constructing the reals with Cauchy sequences. Also, I believe a lot of the constructions of the commonly used examples, such as square root of 2, is easier with Cauchy sequences then with Dekekind- but I don't know this for certain since I've never really used Dedekind cuts, I've just have a rough idea how they work.

By the way- In your Dedekind cut definition you've covered the "closed downward" requirement but haven't you missed "no greatest element" requirement?

Also, if you are claiming the integer 1 and rational 1/1 and real 0.9999.. are the same set then this is wrong! (I'm not exactly clear what your claiming in that line.) However, it is true that the real number that the rational 1/1 homomorphically maps to is the same real represented by 0.999... (Is this what you are saying?)

Dear Thinkabit,

In my first year of university, many years ago, we studied Dedekind cuts and yes, I missed the "no greatest element" condition. We then had it as exercise to prove that Dedekind cuts carry all the field and ordering properties of rational numbers, which allowed us to seamlessly map the rational numbers onto the real numbers.

According to this mapping, "1/1" is mapped to the Dedekind cut containing all rational numbers that are less than 1. Next we can also map all countable-infinite decimal fractions onto Dedekind cuts and if we do so for "0.99999..." then it is easy to show that no elements occur in "1/1" that do not occur in "0.99999..." and vice-versa, thus the two sets are identical, one and the same.

This was in algebra class, independent of the the study of Cauchy sequences in "infinitesimal" class.

wow, thnkabit, I must congratulate you – your obfuscation skills are truly astounding!

Nevertheless, your argument is wrong. You seem to be relying on the false assumption that a number constructed one way can't be the same as a number constructed another way. But in reality there is usually, if not always, infinite scope for needlessly complicating mathematics by including unnecessary trivialities!

'Tis a pity that while you were checking Wikipedia you forgot to check the Integer page http://en.wikipedia.org/wiki/Integer which concurs with my earlier claims.

BTW I'm surprised nobody pointed out the mistake in my last line yesterday. I should have said "parseFloat" as parseInt would be unsuitable for comparison unless I knew the number I wanted translated into integer format was already mathematically an integer.

All logic is not equal. For instance let's look at two logical sequences of numbers. An infinite set of numbers ordered numerically, (... -3,-2,-1,0,1,2,3 ...), then compare that to the same numbers ordered alphabetically. Not as easy to order in that fashion because it's not about counting and knowing which value is greater or lesser numerically, but is reordered based on a different form of logic based on language and which letter comes after an other in the alphabet.

These two forms of logic are not equal, and when applied outside of the scope that that logic is used for makes one set of logic illogical.

For instance in the numerically ordered set the numbers greater then other numbers are the larger one thus the last numbers in this sequence would be greatest numbers in the sequence, the lesser numbers both the earlier numbers, and zero is in the middle of this infinite set of numbers.

Alphabetically, 5 (five) comes before 3 (three) because 5 starts with an "F" while 3 starts with a "T." All negative numbers would be placed in the "N" section of the sequence between "million" and "one." (Assuming you represent million in that way instead of "one-million," or even "a million"). With this sequence and logic Zero is one of the last numbers in this infinite sequence followed only by "zillion" which isn't a qualifiable number just an expression of a very high number.

Take the logic of greater and lesser applied to first and last of a numerical sequence, and try to apply the same logic to numbers alphabetically and it is illogical and no longer valid math or valid logic. Three being later in the sequence is not greater then five.

Not all logic is equal and therefore interchangeable as it is with equations equaling each other are interchangeable in mathematical logic.

he paper proves

1 is a finite number it stops
A finite decimal is one that stops, like 0.157
A non-finite decimal like 0.999... does not stop
when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction

another way

1 is an integer a whole number
0.888... is a non-integer it is not a whole number
0.999... is a non-integer not a whole number
when a integer 1 =a non-integer 0.999... maths ends in contradiction

Dear Sara,

"1" is only a representation, not a number.

It can be designated to represent an integer or it can be designated to represent a real number.

"0.999..." is a common representation of a real number. Again, it is not itself a real number, only a representation thereof.

When "1" represents a real number, it represents the same real number that "0.999..." represents. These are just two different ways of writing the same.

Any number multiplied by "1" remains the same.
Similarly any number multiplied by "0.999..." remains the same.

Similarly, any number multiplied by "8/9" and then divided by "0.888..." remains the same: "8/9" and "0.888..." represent the same real number.

this is waaaaaay out of my comprehension (only went to year 8 primary school then off to work it was at 14 until 54 years later, last year) but in my life it was the decimal . that made all the difference.

sara242,
Restating what the paper purports to prove, and falsely claiming the paper proves it, does not make your argument valid.

Even if you use the non-standard term "non finite decimal" to mean an infinitely recurring representation of a rational number, IT IS STILL A FINITE NUMBER. Indeed even irrational numbers are finite despite the impossibility of representing their value with absolute precision. Only numbers with infinite value are not finite.

0.999... is the decimal representation of the fraction 9/9 (or 3/3 or 6/6 or 7/7 or 11/11 etc) and is equal to 1.
If you read http://en.wikipedia.org/wiki/Integer you'll see that they're defined by how they CAN BE written rather than how they ARE written. As 0.999... can be written as 1 it is an integer.

There is no contradiction and maths keeps going.

To Sara. The paper doesn't prove that different opinions and different logic are equal to eachother. But it does give a few different points to consider with math.

In math equal means that one side of the equal sign is equal invent and interchangeable with with the other side. For instance 5+7 equals and is interchangeable with 3+9. [5+7=3+9].

However not all forms of logic are interchangeable, nor all opinions interchangeable. They are not equal. Even within math there are different methods of logic that are not interchangeable. The example that 1+1=2 and 1+1=1 is a method of two different conclusions based on different logic applied. (One is an equasion with a vaule of adding two other numbers, the other is ignoring the numbers and saying that a number plus another number still equals only one number. The second reasoning is not accurately represented by 1+1=1 to show it's intent. The logic is not equally represented.

As for the 0.9999=1, that math logic is something that is above my knowledge. But if it is true, it is not applicable to other firms of logic. And perhaps even that math logic used two prove 10x=9x is only applicable in theory but not of value in practical use, thus not truly equal or interchangeable to the degree that it represents in the proof.

It was an intresting consideration, but it is not enough to say that all opinions are interchangeable.

Fact is
1 is a finite number it stops
A finite decimal is one that stops, like 0.157
A non-finite decimal like 0.999... does not stop
when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction

another way
1 is an integer a whole number
0.888... is a non-integer it is not a whole number
0.999... is a non-integer not a whole number
when a integer 1 =a non-integer 0.999... maths ends in contradiction

sara242,
Repeating your refuted claims doesn't make them true. They are not valid and never will be. I have already explained why what you label FACT is actually LIES.

I know you want maths to end in a contradiction. What I don't know is why. Is is just that you hate maths?

//Fact is
1 is a finite number it stops
A finite decimal is one that stops, like 0.157
A non-finite decimal like 0.999... does not stop
when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction

another way
1 is an integer a whole number
0.888... is a non-integer it is not a whole number
0.999... is a non-integer not a whole number
when a integer 1 =a non-integer 0.999... maths ends in contradiction//

Yeah... you've said already. Somebody reach over and give sara242 a thump, will ya? She seems to have got stuck.

Fact is the technology you keep repeating yourself relies on maths... and it works. Call off the alarms, folks, and dance a merry jig. For maths still works, oh frabjous day!

Dear Sara,

«1 is a finite number it stops»

1 is a finite number, but that has nothing to do with the fact that it stops.

«A finite decimal is one that stops, like 0.157»

All decimal numbers with a finite number of digits before the dot are finite, whether or not they stop.

«A non-finite decimal like 0.999... does not stop»

0.999... is finite: it is even less than 2.

«when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction»

Both are finite, so the above does not occur.

«1 is an integer a whole number»

Correct.

«0.888... is a non-integer it is not a whole number»

Correct.

«0.999... is a non-integer not a whole number»

Incorrect: 0.999... is an integer. This particular representation may not look like an integer, but it is.

«when a integer 1 =a non-integer 0.999... maths ends in contradiction»

However, "1" and "0.999..." are both integers, in fact they are two different representations of the same integer.

further the paper goes on to prove
1+1=1
contradicting
1+1=2
thus mathematics ends in contradiction

"1+1=2"

Also:
1+1=11 (side by side)

1+1=4 (rotate one '1' counter-clockwise)

1+1=8 (2 cats produce 6 kittens)

The outside world of shapes and forms is full of contradictions because it is not (and cannot be) well defined, in fact it is an illusion.

However, mathematical objects and the operations that apply to them are well defined so the results of these operations are always the same. Similarly, a geometric point has no shape, form or size, so nobody can draw a geometric point in the world that is outside mathematics.

You seem to mix up the mathematical objects with their representation: a "number" is not a mathematical object, it only represents a mathematical object. Mathematical operations can only be applied to mathematical objects, thus:

1 number + 1 number = 1 number

cannot be reduced to "1+1=1" since you cannot divide (both sides) by "number", which is not a mathematical object.

//further the paper goes on to prove
1+1=1//

Not in any conventional sense of the word 'prove'.

Get off the crack, it's not doing you any favours.

The fact is
The paper proves

1 is a finite number it stops
A finite decimal is one that stops, like 0.157
A non-finite decimal like 0.999... does not stop
when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction

another way

1 is an integer a whole number
0.888... is a non-integer it is not a whole number
0.999... is a non-integer not a whole number
when a integer 1 =a non-integer 0.999... maths ends in contradiction

and

1+1=2
and
1+1=1
1 number +1 number=1 number
or
1 heap of salt + 1 heap of salt= 1 heap of salt
thus maths ends in contradiction

Dear Sara,

One cannot perform arithmetic operations on "a heap of salt", only on well and axiomatically-defined numbers.

AND

0.999... IS an integer. Even the paper you presented does not claim that it isn't, how less so proves so.

when a finite number 1 = a non-finite number 0.999.., this simply demonstrates that one number can be represented in two (or more) different ways. In this case, one WAY OF REPRESENTATION is finite, the other not: that says nothing about the finiteness or otherwise of the actual number itself.

sara242,
The sum "1+1=1" is never right. We can make it right by multiplying each side be zero, but mathematicians are well aware of that, and multiplying both sides by zero is not treated as equivalent to multiplying them by another number because we know from observation that it makes things equal that would ordinarily be unequal. When an apparent contradiction is encountered, mathematics sets conditions to avoid it; the contradiction is resolved and maths keeps going. This is because mathematics is about truth - and the truth is that 1 is not equal to 2 even if you "prove" otherwise by multiplying by zero.

Zero isn't the only value which has to be treated differently in some mathematical laws. The same goes for infinity, and for minus infinity. NonSpecificNumber appears to be a fourth such value, though many would dispute its validity as a value. Likewise with a heap of salt.

Either way, mathematics sometimes has exceptions, but it never ever has contradictions (except as proof that something is FALSE). If you think you find a contradiction, it shows that either your reasoning is wrong or your assumptions are wrong (or both).

1+1=2 no matter how much you want it to be something else.

Aidan, Yuyutsu... I reckon it might be time to stop feeding the troll, fellas. She's clearly just hungry for attention.

In the paper it is proven by logic that when mathematics/science ends in contradiction you can prove anything you want in mathematics ie you can prove Fermat's last theorem and you can disprove Fermats last theorem

It should also be born in ind that
The paper also proves in logic that all opinions are equally valid ie feminism is valid and anti-feminism is valid. Also the opinions for gay marriage are valid: for gay marriage and against gay marriage are equally valid

Ok Sara. Let's look at this from where you want it to go. Give it a chance.

If all opinions are equal, where do you go from there. For arguments sake, let's ignore that we disagree with the origional point. It sounds like an opening point to lead to another. What's the point you want to address from there?

Where do you go when all opinions are equal?

For me, I have come to terms of some things being right and others not. That being trained by those who know a skill, makes their judgment on that trade valuable. Their opinions greater then those who know nothing of that skill. I have also seen both lies and truths fight for a spotlight. Knowing that the truth is greater then fiction is. Even to say that one way of life can be harmful, and another healthy, so listening to those who have lived longer is also of more value because they can see past the philosophy and see what was harmful and what was healthy.

I tell you this so that you know, we will not agree with the first point, that all opinions are equal. But for arguments sake let's ignore that. If all opinions are equal, where do you go from there?

All opinions for and against something are equally valid

All things are possible

Now with the inconsistency of language
all possible views and their negation are
now possible and equally valid Thus the
philosophies of Kant Hegal Plato
Aristotle etc all philosophies and the
negation/opposite of the philosophies of
Kant Hegal Plato Aristotle etc all
philosophies are now possible and equally
valid

and

in the every day world this means that all
views are valid but so are the opposing
views valid Thus all civil rights views are
valid ie pro gay marriage is valid but so is
the opposing view ie anti-gay marriage is
valid
as
Each view contains within it its negation
as all views end in meaninglessness

Thus
We can now just treat all views/ philosophies esthetically that is for their logical and argumentative beauty rather than for any scientific or truth value just like one treats poetry painting music for their esthetic beauty

Ok, so first you don't need math condrictions to make the point that there can be beauty in any philosophy. You don't even need to say that those philosophies have validity, or even that they are true.

However, that point that there is beauty in any philosophy isn't always true.

That said. The undertone of your comment should be worth discussing. Being accepting to any and all philosophies.

There's a difference between be accepting and being loving. Some things are rejected purely because they are unhealthy. It's a parent's job to steer their children into healthy attitudes and knowing the truths of the world, so to prepare them for adulthood and help them grow. It also means trying to help them out of negitive philosophies and negitive influences.

For example anorexic philosophies are unhealthy and harmful. There is no beauty in thinking that starving yourself makes you beautiful. It is not equally valid as other philosophies.

To Sara.

I'm leaning on the understanding that your view of everything being equally valid is out of wanting to be loving and accepting. If that understanding is wrong let me know. The following is based on that understanding.

There's a philosophy I've grown up with known by the phrase "love the sinner, hate the sin."

The idea is that we can be loving without accepting other people's flaws as OK. Accept them as much as you can, but don't accept a philosophy that is harmful, or a behavior that is wrong.

It gets deeper from there. The idea that you accept others in spite of their flaws, has a benefit because no one is perfect. Everyone has something they can work on, something they can correct, probably even something that is wrong, harmful or somehow unhealthy. Have mercy on others because you also want mercy. Same with forgiveness. Even to say that it takes a true friend to help another out from continuing a bad path, (such as taking away their car keys when a friend is drunk).

You can accept and love others without accepting their wrongs.

The paper proves

1 is a finite number it stops
A finite decimal is one that stops, like 0.157
A non-finite decimal like 0.999... does not stop
when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction

another way

1 is an integer a whole number
0.888... is a non-integer it is not a whole number
0.999... is a non-integer not a whole number
when a integer 1 =a non-integer 0.999... maths ends in contradiction

At this stage sara242 due to your repeated claims that have been refuted repeatedly I think you are either just a troll for fun or you simply don't understand much about the low foundational levels of maths.

The explanation that I gave you before is 100% valid when you treat the set of integers as a separate object that is not a subset of the reals (but which however may be homomorphically mapped into the reals, ie: for each integer there corresponds a real and the arithmetic operations are preserved, eg: 1-maps_to->1.000.., 2->2.000.. etc where 1+1=2 and 1.000+1.000=2.000, 2*2=4 and 2.000*2.000=4.000, etc) . To recap what I said: since an integer cannot ever equal a real the premise to "when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction"is always false thus this statement (although overall true) does not by itself allow you to claim the conclusion is true.

If however you are of the opinion that the integers should be considered a subset of the reals the argument still fails because by all the methods used to construct real numbers (Cauchy sequence, Dedekind cuts, Tarski axioms, etc) the integer 1 (which is now actually the real 1.000..) is the same number represented by 0.9999...

Note: technically you are free to chose to treat integers as a separate set or as a subset of the reals- at the end of the day your maths will still work. However, personally I'm adamant that you should keep them separate for reasons of philosophy, practicality, efficiency and history.

By-the-way: the number 0.999.. is a finite number but I assume that you mean the representation is not finite or the process to construct it is not finite or the set that it actually is has infinite members.

What the opinion means about all opinons being valid is

in the every day world this means that all
views are valid but so are the opposing
views valid Thus all civil rights views are
valid ie pro gay marriage is valid but so is
the opposing view ie anti-gay marriage is
valid
as Each view contains within it its negation
as all views end in meaninglessness

also

Now with the inconsistency of language
all possible views and their negation are
now possible and equally valid Thus the
philosophies of Kant Hegal Plato
Aristotle etc all philosophies and the
negation/opposite of the philosophies of
Kant Hegal Plato Aristotle etc all
philosophies are now possible and equally
valid

plus

The paper proves

1 is a finite number it stops
A finite decimal is one that stops, like 0.157
A non-finite decimal like 0.999... does not stop
when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction

another way

1 is an integer a whole number
0.888... is a non-integer it is not a whole number
0.999... is a non-integer not a whole number
when a integer 1 =a non-integer 0.999... maths ends in contradiction

If you repeat a given statement a sufficient number of times, then eventually it will become true.

The catch is, that number may be very big so you might not live long enough to repeat the statement that many times!

There will indeed come a time when all views and their opposites will be equally valid - that is, when the oceans boil as the sun turns into a red giant and no humans remain, thus no views either.

Ok sara, it's obvious that you refuse to even try to understand the flaws in your maths. So let's try another approach.

Did you know that back in the 1890's set theory as then defined/used was found to contain an inconsistency known has Russell's paradox (Russell's found it in 1901 but it was discovered earlier by others). This paradox showed that naive set theory was inconsistent, so for those who used it as a base starting point for maths had to accept the 1+1=3 and any and all other other math statements if they applied the classical logic "principle of explosion" (or in fancy latin "ex contradictione sequitur quodlibet").

But guess what- this revelation that maths was inconsistent didn't change the fact the sun rose the day before this contradiction was found, it didn't change the fact that opposite poles of a magnet attract nor the fact that the a native born Englishman will likely be able to speak English while a Japanese person speak Japanese. Indeed, the only "truths" that it did change were those directly concerning or built upon math.

The point is: So what if maths is inconsistent? This only effects the ability for us to do maths sensibly. We still exist and we are still able to get factual answers to questions that don't involve math by physical empirical observation of the world around us. (eg: facts like the sky is blue, I don't like to experience pain or abuse, animals need to eat, etc.) We can even determine real world facts and compare if they much a particular person's opinion.

By-the-way: when the mathematicians/philosophers did find Russell's paradox, instead of claiming all sorts of weird math results like 1+1=3 all they did was change the underlying foundations so that the weirdness went away: ways they did this was by creating a better set theory (eg: ZFC) and another by was by inventing type theory and introducing a system of types. The world didn't end and not every person's opinion was true.