Questions tagged [peano-axioms]
For questions on Peano axioms, a set of axioms for the natural numbers.
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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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How do we define the natural numbers through Peano axiomatics if they are already defined through predicates axioms? [closed]
First of axioms Peano - "0 is a natural number."
That axiom is predicate.
According to Gottlob Frege, the meaning of a predicate is exactly a function from the domain of objects to the truth-...
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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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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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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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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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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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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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Why is addition not completely defined here?
Say for the natural numbers, we define addition this way:
$0 + 0 = 0$, and if $n+m = x$, then $S(n) + m = n+S(m) = S(x) $
Say we have the regular Peano axioms, except we delete the axiom 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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Is it circular to include reachability from $0$ like this as a Peano axiom?
I am wondering whether it makes logical and semantic sense to include, as an axiom to define the natural numbers, that "every natural number is either $0$ or the result of potentially repeated ...
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Specific example of a property $P$ that Peano arithmetic proves holds true for every specific number, but not for all numbers.
Can someone give a specific example, if there is any, of a predicate $P(x)$ expressible in the language of Peano arithmetic, such that the first-order theory of Peano Arithmetic proves $P(0)$, $P(1)$, ...
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