Re: Axiomatization of Ordinal Arithmetic


Peter_Smith wrote:
Charlie-Boo wrote:
Peter_Smith wrote:
Charlie-Boo wrote:

It's much easier (better by Occam's Razor) to see that Peano's Axioms
amount to the assertion that the set of natural numbers is recursively
enuerable. Do you agree to this equivalence?

When you say that one claim (or set of claims) "amounts to" another
claim, that is a naturally read as an equivalence claim. I pointed out
that, quite trivially, to say that PA applies to the X's and to say
that the X's are r.e. are not equivalent claims (there are instances of
X for which the first is false and the second true).

Give an example or admit that you're a liar (the only honorable
alternative.)

C-B

I see that you are trying, in your usual inimitable way,

Is that good?

to raise the
tone of the debate here.

I gave an example. Saying the naturals are r.e. is consistent with
there only being finitely many of them, while Peano's Axioms aren't
consistent with there only being finitely many of them.

Good point. Make that infinite and r.e. My point is that the program:

1. X = 0
2. Write X
3. change X to X+1
4. Go to (2)

enumerates the natural numbers if Peano's Axioms are true, and if it
enumerates the natural numbers then Peano's axioms are true. However,
Peano's axioms make sure that the set of X enumerated is infinite, so
add that condition.

C-B

So Peano's
Axioms can't amount just to the assertion that the set of natural
numbers is recursively enumerable.

.

Relevant Pages

  • Re: Axiomatization of Ordinal Arithmetic
    ... that is a naturally read as an equivalence claim. ... Saying the naturals are r.e. is consistent with ... Axioms can't amount just to the assertion that the set of natural ...
    (sci.logic)
  • Re: infinity
    ... >> Not while discussing the consequences of the axioms as they ... I commented that they were irrelevant to, and contradictory ... existed for over 2 millennia without needing any infinite naturals. ... A natural number is finite if the set of naturals less than or equal to ...
    (sci.math)
  • Re: Calculus XOR Probability
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    (sci.math)
  • Re: help with Godels
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  • Re: Cantors circular "proof" that evens = integers
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