Re: Peano's Axioms are Inconsistent

On Aug 12, 5:26 pm, A Nony Mouse <T...@xxxxxxxxxx> wrote:
In article

And that is the crux of the matter. You can count all the natural
numbers (that is the definition of countable, after all), but at no
point will you have counted all of them.

You can count all the natural numbers and you can't count
all the natural numbers? This sounds like a contradiction to me.

The entire essence of the necessity for distinction is that the size of
the set of natural numbers is not a natural number itself.

Yes, the size of the natural numbers, as defined by Peano's Axioms,
is a natural number.

The Peano axioms say nothing at all about the sizes of any sets,
collections, aggregates, assemblages or whatever.
In some versions, including the original Peano version, they do not
mention sets at all.

There are no sets.

 I prove this to be true.

Then you are claiming to have proved a lie.

My mistake. I shouldn't assume there are sets.

You assume PA is consistent.

We only assume that RussellE is inconsistent. At the moment that is our
only axiom. And it ceratainly qualifies as self evident.

Let's see what we know about your formal theory.

Sets don't exist because PA doesn't define sets.

If sets did exist, you can't use them in proofs
in set theory.

The word "all" doesn't really mean all.

PA doesn't define the size of sets because
sets don't exist and the natural numbers
have nothing to do with size.

Strings like "AxEy T(y)(x) = '1' " don't mean anything
because "Ax" is defined as something that
punches holes in leather.

And your only axiom is "RussellE is inconsistent".
Oh, and the Axiom of Ambiguity.

Good luck with that formal system.


Russell
- Integers are an illusion
.

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