Re: Peano axioms
- From: José Carlos Santos <jcsantos@xxxxxxxx>
- Date: Wed, 17 Oct 2007 14:02:16 +0100
On 17-10-2007 11:32, dwwdkddb wrote:
A couple of questions about the Peano axioms:
Interesting. You state that you are going to ask us a "couple of
questions" and you ask three questions. It becomes more interesting if
we take into account that the subject of these questions are the Peano
axioms.
1) In the original formulation of the axioms, Peano included four
axioms concerning the equality (=) relation. Why are these normally
left out nowadays?
2) Why isn't it an axiom that the (set of) natural numbers actually
exist(s)? How is this obvious?
It is not obvious. But this is how axiomatic systems are: they consist
of a list of objects together with a list of relations between them. The
existence of objects satisfying these relations is a question to be
addressed outside that system.
3) When writing the axiom that 0 (or 1) is a natural number in formal
notation, is it enough to write "1 \in \mathbb{N}", or should it be
"\exists 1\in\mathbb{N}"?
The first option is correct; the other one isn't. The objects of the
Peano axioms are three: N, 1, and _s_. The first axiom states that 1 is
an element of N.
Best regards,
Jose Carlos Santos
.
- Follow-Ups:
- Re: Peano axioms
- From: dwwdkddb
- Re: Peano axioms
- From: Helmut Richter
- Re: Peano axioms
- References:
- Peano axioms
- From: dwwdkddb
- Peano axioms
- Prev by Date: Re: Implementable Set Theory and Consistency of ZFC
- Next by Date: Re: negative base raised to fractional exponent
- Previous by thread: Re: Peano axioms
- Next by thread: Re: Peano axioms
- Index(es):
Relevant Pages
|
