All Questions
Tagged with peano-axioms foundations
54
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Allegedly: the existence of a natural number and successors does not imply, without the Axiom of Infinity, the existence of an infinite set.
The Claim:
From a conversation on Twitter, from someone whom I shall keep anonymous (pronouns he/him though), it was claimed:
[T]he existence of natural numbers and the fact that given a natural ...
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Why Löwenheim–Skolem theorem asserts the non-existence of such predicates in 1st order logic
Suppose there was a predicate, in the language of 1st order $ \mathsf {PA} $, such that it is only true for standard natural numbers i.e. it accepts ALL and ONLY standard natural number, and it ...
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Confusion about $\mathsf{PA}$'s self-provable consistency sentences
Edit: This long question was basically answered by a quick comment! I'll accept an answer if someone posts one, but it's basically answered already.
Background:
In Peter Smith's Introduction to Gödel'...
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On the Axiomatic Foundation of Elementary Number Theory
I am under the impression that there is a set of axioms on which elementary number theory unfolds. If this is true, what are the axioms? Are they the five Peano axioms (at least, thought there were ...
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How do Robinson arithmetics axioms prevent this model of N?
In the Peano arithmetic wikipedia article, a figure is shown on why the axiom of induction is necessary, without it, the set of whole dominos would be a valid representation of N
In Robinson ...
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Can the subsitution property be deduced from the peano axioms?
How does the substitution property of equality, i.e. for any function on the naturals $f : \mathbb{N} \to \mathbb{N}$,
$$ \forall a \in \mathbb{N} . \: \forall b \in \mathbb{N} . \; a = b \implies f(a)...
- 29
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Presburger arithmetic is consistent, but relative to what?
In the Wikipedia article for (the first-order theory of) Presburger arithmetic, it is stated (among other properties) that Presburger arithmetic is consistent. What meta-theory does he rely on in ...
user923098
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What proof system is this paper by Hamano and Okada introducing?
In Hamano and Okada's "A relationship among Gentzen's proof-reduction, Kirby-Paris' hydra game, and Buchholz's hydra game", they briefly introduce a proof calculus for Peano arithmetic, but ...
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Godelian sentences in other first order languages
I've been asked to teach an intro to logic and computation course. This is not a field that I am particularly familiar with, but I am happy to have the chance to learn it - hopefully - properly.
Since ...
user923098
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Implications of $Q\vdash \lnot Con(Q)$
By Gödel's incompleteness (which holds even for $Q$ as discussed here) we have that the Robinson arithmetic $Q$ is consistent iff $Q+\lnot Con(Q)$ is consistent.
As I understand it, without further ...
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How to justify the necessity of the Axioms?
I am studying the logical fundaments of mathematics, but very often I have trouble to understand Peano's and ZFC/ZFC axioms.
In Tao's book Analysis I, I found very helpful when he points out what ...
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What is the difference between "Peano arithmetic," "second-order arithmetic," and "second-order Peano arithmetic?"
I think this needs to be clarified, so it would be helpful to see an answer to this somewhere.
I've seen the following terms:
Peano arithmetic.
Second-order arithmetic.
Second-order Peano arithmetic.
...
- 2,461
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Are the naturals really a subset of the real numbers? [duplicate]
Ok, so this question seems obvious, right?
But what I mean is the numbers in the way they are logically / axiomaticaly defined in the foundations of Mathematics. As far as I know, the naturals are in ...
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Peano axioms set theory
Arithmetic is the core of the study of numbers; from natural numbers you can construct the other numbers. Now, I have heard that set theory is the most fundamental core of mathematics; for this reason ...
- 369
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Understanding the natural numbers and Peano's axioms
I am reading the book Analysis $I$ by Terence Tao, where the following is written: [...] $\mathbf{N}$ should consist of $0$ and everything which can be obtained from incrementing: $\mathbf{N}$ should ...
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