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.