Re: Peano's Axioms are Inconsistent

RussellE wrote:
On Aug 11, 6:50 pm, Joshua Cranmer <Pidgeo...@xxxxxxxxxxxxxxx> wrote:
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. I've already described why that implies the distinction, but you seem to have been unable to understand it then and I see no reason why you will be able to understand it now.

Including the last 100 blank positions at the end to the tape.

There is no last position, as there is no largest natural number. Your only evidence for this is an argument using natural (English) language, which is ambiguous in many cases. Note especially that the concept of "all" in English predates the understanding of the finer points of "infinity" (which itself came around only in the 1800s or thereabouts, I think)--extrapolation is always a risky adventure, and ambiguous language even more so.

The old Axiom of Deniability. You simply deny there is
a contradiction. By assuming PA is consistent, you can
"prove" there is no paradox.

Your proof for the contradiction relies entirely on a colloquial usage of the word "all." Colloquial usages are often different from mathematical usage; sometimes, some subfields even make distinctions that others do not. For example, a circle can be considered to be a sphere in some areas.

At best, you have evidence that letting aleph-null be a natural number creates a contradiction, but since no one excepts you takes that as an axiom, it's completely contrary to proving the inconsistency of Peano's Axioms.
--
Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth
.

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