All Questions
Tagged with peano-axioms arithmetic
63
questions
2
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72
views
Is there a problem if I don't use $0$ in Peano arithmetic?
Peano arithmetic is the following list of axioms (along with the usual axioms of equality) plus induction schema.
$\forall x \ (0 \neq S ( x ))$
$\forall x, y \ (S( x ) = S( y ) \Rightarrow x = y)$
...
- 103
0
votes
1
answer
261
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Peano’s Fifth axiom as stated by Russell
I’m translating Russell’s Introduction to Mathematical Philosophy. I’m having difficulty understanding his formulation of the fifth axiom:
(5) Any property which belongs to 0, and also to the ...
- 437
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1
answer
160
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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 ...
- 3,219
2
votes
1
answer
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Mistake in proof that $\textbf{P}^- +\textbf{I}\Delta^0_0$ proves infinitude of primes
I know that it is still an open question whether $\textbf{P}^- +\textbf{I}\Delta^0_0$ is able to prove that there are infinitely many primes. I know that $\textbf{P}^- +\textbf{I}Open$ cannot prove ...
- 1,926
4
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2
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523
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How do Peano's axioms make it clear what the successor is equal to?
Most likely this subject has been covered many times here, still I fail to grasp this.
I can't understand how do we know that the successor of $1$ is $2$ based on Peano's axioms, given that we start ...
- 1,091
1
vote
1
answer
183
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What ZF can do and Peano's axiom cannot.
I am interested in how much math can be done from Peano's axioms and what can't.
What is there in the mathematics done with ZF that cannot be done with Peano's axioms?
- 390
2
votes
2
answers
111
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First order theories stronger than PA
I'm trying to learn basic logic, and it seems that Peano Arithmetic — a first order theory of the natural integers with $+$ and $\times$ symbols, is a common object of interest.
We could define (a ...
- 785
1
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1
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288
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Can Goodstein's theorem be false in some model of Peano arithmetic? How?
I read that Goodstein's theorem is not provable in Peano arithmetic, but is provable in ZFC.
Now, using Godel's completeness theorem, we can say that, since Goodstein's theorem is not provable in ...
- 1,437
1
vote
6
answers
322
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In the Peano Axioms, how do we know that the successor of a natural number $n$ is $n + 1$? (It's never explicitly stated that it should be "$+1$")
The Peano Axioms never explicitly state that the successor function is $n$ $+ 1$. Is this just taken by convention? It's as if we should already know what the natural numbers are and know that the ...
- 197
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0
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74
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How can one understand the axiomatization of numbers and their operations?
I've recently taken an interest in arithmetic and how it can be axiomatized. I haven't been reading very deeply about some of the known axioms I've come across, like Peano axioms, which by my ...
- 57
1
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0
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251
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Induction with an existential quantifier
I am looking at an altered condition in the standard natural number induction principle (see reminder below). Specifically, the schema
$$\forall n. \Big(\big(n=0\lor \exists p. ((Sp=n)\land \phi(p))\...
- 12.3k
2
votes
0
answers
57
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A Peano system with an infinite initial segment
Let $T$ be a binary tree with the lexicographic order. And $f:T→T$ be the successor operation.
We denote the empty sequence in $T$ by $i$.
Also Suppose that:
For all $X⊂T$ if these two conditions hold:...
1
vote
0
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116
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Peano arithmetic without addition
Take the usual theory of First Order Peano Arithmetic with the signature $(0, s, +, \times)$. Now consider these two questions:
Take the set of PA theorems which don't involve '$\times$'. Is there a ...
- 55.4k
0
votes
1
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Can I use a addition table with infinite length and height to define addition on the natural numbers rather than Peano's Axioms? [closed]
I'm reading David Steward's "Foundations of Mathematics" and in chapter 8 he is building an axiomatic system for the natural numbers with addition defined using Peano's Axioms. I don't ...
- 3
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1
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Can Peano arithmetic prove the consistency of "baby arithmetic"?
I am reading Peter Smith's An Introduction to Gödel's Theorems. In chapter 10, he defines "baby arithmetic" $\mathsf{BA}$ to be the zeroth-order version of Peano arithmetic ($\mathsf{PA}$) ...
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