Questions tagged [peano-axioms]
For questions on Peano axioms, a set of axioms for the natural numbers.
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Godel's incompleteness theorem: Question about effective axiomatization
I am studying Godel incompleteness theorems and I am struggling with the definition of effective axiomatization.
From Wikipedia:
A formal system is said to be effectively axiomatized (also called
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Can we modify the Peano axioms like this? [closed]
I am wondering if the following modifications of the Peano axioms result in a set of axioms equivalent to the Peano axioms, in the sense that any set of numbers satisfies these modified axioms if and ...
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Understanding Peano Arithmatic and Axioms
I am new to analysis and started reading a PDF I found on Reddit, the link is here.
I stumbled on a few question about basic Peano axioms and the definitions that the PDF derived from it.
In case ...
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Proving the existence of hyperoperations in a Peano system
In Mendelson's "Number Systems" he defines and subsequently proves the existence and uniqueness of binary operators $+$ and $\times$, as well as exponentiation, in $P$ using the so-called ...
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Why is the material conditional treated like logical entailment in second order quantification? [closed]
According to this Wikipedia article the second order axiom of induction is: $$\forall P(P(0)\land \forall k(P(k) \to P(k+1)) \to \forall x(N(x) \to P(x))$$
Where N(x) means x is a natural number. That ...
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Finite axiomatization of EFA
According to the paper Fragments of Peano's Arithmetic and the MRDP theorem (Section 6), elementary function arithmetic (EFA) is finitely axiomatizable.
Is there a known explicite finite ...
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Naive Set Theory - proof of commutativity of products
I am working through Halmos's Naive Set Theory on my own and trying to do all the exercises, including what are merely suggestions in the text. I am right now in section 13, which shows a derivation ...
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Axiomatic reason why $a=4 \implies a>1$ for $a \in \mathbb{N}$
This is a trivial task:
Given $a \in \mathbb{N}$ and $$a=4$$
Show $$a > 1$$
Part of the challenge for newcomers like me is that "easy" tasks actually make it harder to think about the ...
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Ability of Peano axiom with integer set?
Axioms:
Peano Axioms (defines natural number, introducing 0 and ')
For each predicate φ, there exist exactly one set X, s.t. forall x, φ(x) <=> x∈X.
So, it's possible to define less-than in ...
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What inference rules do people refer to when they talk about "the inference rules" of Peano Arithmetic? [duplicate]
There are many examples in the literature (for example, in this question)
where the author says that something "is provable in Peano arithmetic" or "is not provable in Peano arithmetic&...
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Proof that each natural number has a unique successor
I've proven that every positive natural number has a unique predecessor using Peano's axioms. But now, I was wondering how I could prove that every natural number has a unique successor using the same ...
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not precisely understanding what is asked by ex 3.5.13 Tao Analysis I (only one natural number system)
I am failing to understand the intent of the question posed by exercise 3.5.13 of Tao's Analysis I 4th ed.
The purpose of this exercise is to show that there is essentially only one version of the ...
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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 ...
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How to justify why succession and addition cannot be circularly defined like this?
I am reading Tao's Analysis I, in which he states:
One
may be tempted to [define the successor of $n$ as] $n + 1$. . . but this would introduce a circularity in our foundations, since the notion of ...
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Question about the Peano axioms + linear order axioms
The signature consists of $S$, $0$ & $<$ and the axioms are:
I - $\forall x (S(x) \not= 0)$
II - $\forall x \forall y (S(x) = S(y) \to x = y)$
III - First-order Induction schema
IV - $<$ is ...
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