Questions tagged [peano-axioms]
For questions on Peano axioms, a set of axioms for the natural numbers.
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How is exponentiation defined in Peano arithmetic?
How would exponentiation be defined in Peano arithmetic? Unless $n$ is fixed natural number, $x^n$ seems to be hard to define.
Edit 2: So, what would be the way to define $x^n+y^n = z^n$ using $\...
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How does (ZFC-Infinity+"There is no infinite set") compare with PA?
How does (ZFC-Infinity+"There is no infinite set") compare with (first order) PA? Intuitively, neither theory should be more powerful than the other.
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A question on Terence Tao's representation of Peano Axioms
While reading Terence Tao's book on Analysis I had some questions regarding the implication of the Peano Axioms.
After writing the following four axioms (which I will write without changing their ...
user170039
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Presburger arithmetic
In discovering that Presburger's arithmetic is one of the weaker systems in PA that does not violate Godel's first incompleteness theorem. Upon reading the wiki article, it said that Presburger proved ...
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Why is the Axiom of Infinity necessary?
I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
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What is an example of a non standard model of Peano Arithmetic?
According to here, there is the "standard" model of Peano Arithmetic. This is defined as $0,1,2,...$ in the usual sense. What would be an example of a nonstandard model of Peano Arithmetic? What would ...
user223391
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Deducing PA's axioms in ZFC
Recently I've stumbled across this claim:
Peano axioms can be deduced in ZFC
I found a lot of info regarding this claim (e.g. what would (one version of) the natural numbers look like within the ...
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Prove that no positive integer is both even and odd, and that all positive integers are either even or odd
What is says on the can:
Prove that no positive integer is both even and odd, and that all positive integers are either even or odd.
This, of course,
depends on defining even and odd.
For extra ...
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Purpose of the Peano Axioms
Wikipedia says the Peano Axioms are a set of axioms for the natural numbers. Is the purpose of the axioms to create a base on which we can build the rest of mathematicas formally?
If this is true ...
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Why are addition and multiplication included in the signature of first-order Peano arithmetic?
In the second-order approach to Peano Arithmetic, the only non-logical symbols are the constant $0$ and the successor function $S(*).$ But, when we go to first-order Peano Arithmetic, something goes ...
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Growth-rate vs totality
How can one prove the statement, "If a function grows fast enough, it cant be proven total in PA, unless PA is inconsistent"? How fast must it grow to be not provably total?
user77335
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Are the standard natural numbers an outstanding model of PA?
Is the model $\mathcal N$ of the standard natural numbers in any way outstanding from all the possible (non-standard) models of PA? For example, might it be that $\mathcal N$ is some kind of a minimal ...
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Can Peano Arithmetic show that the Continuum Hypothesis is Independent of ZFC?
Can Peano Arithmetic show that the Continuum Hypothesis is Independent of ZFC? In other words, is $PA \vdash Con(ZFC) \implies Con (ZFC + CH) \land Con(ZFC + \lnot CH)$ true?
I believe the answer is ...
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Why does induction only allow numbers connected to $0$ to be natural?
When learning Peano axioms normally we use induction to suggest that if a property $P(0)$ is true, and if $P(k)$ being true implies $P(k + 1)$ being true (for natural number $k$), then if $P$ is the ...
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Proving that Order for N is Anti-symmetric
I'm having trouble deriving the following fact from the basic properties of $+_{\mathbb{N}}$ and the definition:
Definition. $\forall n, m \in \mathbb{N}: (n \geq m) \leftrightarrow (\exists a \in \...
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