Re: continuum hypothesis and 0=1

On 2007-03-04, Pierre-Yves.Gaillard@xxxxxxxxxxxxxxx wrote:
What is the "standard model"?

In this context "standard model" refers to the structure of naturals
together with addition, multiplication and the successor function. When
speaking of the truth or falsity of arithmetic statements nothing mysterious
is going on; "Goldbach's conjecture is true", for example, is equivalent to
"every even natural greater than two is the sum of two primes", and so on.
In most contexts qualifying "true" with "in the standard model" is
pointless, since non-standard models, or models at all, are not considered,
and are indeed entirely irrelevant. Non-standard models of arithmetic and
such like are studied in certain specialized and rather marginal branches
of mathematical logic, and in that context explicitly mentioning when we're
dealing with the standard model might be in order.

Are there mathematical statements which have "nothing to do with any
particular set of axioms"?

Most mathematical statements have nothing to do with any particular set of
axioms.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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