Prove that 0 ≠ 1 for natural numbers 0 and 1. Prove that you

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0 is the identity of addition for natural numbers while 1 is not the identity of addition for natural numbers. This means that 0 has a property that 1 has not. Therefore 0 cannot equal 1. QED

>>8423582

How do you know 1 is not the identity for addition? If 0 = 1, it would be.

Also, even if you answer the first exercise (hint: define a natural number), can you do the second?

X + 0 = X
X + 1 ≠ X

>>8423620

How do you know X + 1 ≠ X (for at least one X)?

You're assuming that one of: 0 ≠ 1, 1 ≠ 2, 2 ≠ 3, ... holds. Your proof is circular, or at least has an unsubstantiated claim.

>Prove that you cannot prove that 0 = 1.**
Impossible

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Second_incompleteness_theorem

>>8423626

So you're saying that you cannot prove that you cannot prove that 0 = 1? This is assuming that our axioms are consistent. They might not be.

If our axioms are inconsistent, you CAN prove that you cannot prove that 0 = 1, (as well as proving that 0 = 1.) That's why it's a two starred problem - it might be possible, nobody knows.

>>8423562
>this is what mathematicians on coffee break do

Now I'm kinda sad I don't know a single pure mathematician.

>>8423634
>it might be possible, nobody knows.
Not that guy, but you're also being extremely sloppy and throw in assertions left and right.
Maybe somebody knows.

1=S(0)
Thus 1 != 0 (if so then we would have 0=S(0))

>>8423562
Using the peano axioms. (Look them up, not going to list them for you)

1 = S(0) by definition.

Now, suppose the contrary. Suppose 0 = 1
0 = 1 implies
0 = S(0)

Lets call this conclusion statement 1.

Also, y = S(x) implies that y has a predecessor, namely x. So going back to statement 1

0 = S(0) implies that 0 has a predecessor, which means that there exists some natural number a such that S(a) = 0 and this contradicts the axiom that says that there exists no natural number such that its successor is 0.

Therefore our assumption that 0 = 1 must be wrong, therefore 0 is not equal to 1.

>>8423562
lel. :P

I'll define "value" as a characteristic that symbols can have. Symbols may have different amounts of values. The amount of values a symbol has an undefined representation and is only used to relate differences in values.

I'll define equality "=" as an operator that does nothing to any values, however, it can either have the characteristic of being "true"(=) or "false"(≠). And it is only true if of the operands that it is between; the one with the less values is its predecessor. Else-wise it is false.

I'll define "0" as a symbol called "natural number" with more values than the symbol "1" which has the same name "natural number."
therefore: 0 ≠ 1

I never defined which direction you read the symbols so 0 ≠ 1 and 0 = 1 both holds given what direction you read the symbols. However, unless the writer tells you what way they are intended to be read. There is no way to tell whether the writer used the symbols correctly.
QED?

>>8423858
eh, you're basically missing out on Big Bang Theory-tier jokes and memes.

>>8423562
prove that you can prove that you cannot prove that 0 = 1
and then maybe we'll go to work
you
are a bastard
and a
faggot

>>8423562
>Prove that you cannot prove that 0 = 1.**
What's meant by **?