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Topic: Infinity: The Story So Far
Replies: 16   Last Post: Mar 7, 2014 8:31 AM

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Peter Percival

Posts: 2,623
Registered: 10/25/10
Re: Infinity: The Story So Far
Posted: Mar 7, 2014 8:31 AM
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Virgil wrote:

> With suitable interpretations of first element and successorship,

That is the crux of the matter! If the first element is zero, and if
successor means add one, then the things that the Peano axioms are about
are indeed the natural numbers. But the 0 and S of the language of
(first order) Peano arithmetic are undefined primitives.

> the
> Peano axioms define every and any inductive set, which either form of |N
> is certainly one of.

Madam Life's a piece in bloom,
Death goes dogging everywhere:
She's the tenant of the room,
He's the ruffian on the stair.

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