All Questions
Tagged with peano-axioms number-theory
25
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Non recursive provably total functions
It is provable that every primitive recursive function is total in 1st order PA. Some non primitive recursive functions are also provably total in PA.
Can we show that a function is totally provable ...
user1203803
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Meaning of Provable recursiveness
Is there any difference between provably total function and provable recursiveness of a function in first order PA ?
From provably total I mean that the totality of the function itself is provable in ...
user1203803
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1
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95
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Why is does this first-order set of axioms NOT genetically define the natural numbers?
It is a theorem of model theory that any recursively enumerable set of axioms $\Gamma$ for number theory permit non-standard models. That is, if there is one model for $\Gamma$, then there are two ...
- 1,093
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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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Why is a unary numeral system bijective?
In this treatment of unary numeral system, it is called "bijective base-1 numeral system." I understand bijective from functions, but why is it used here? Also, the article says the Peano as ...
- 948
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Can this function be described by a formula?
Suppose $PA$ is Peano arithmetic. For $m \in \mathbb{N}$ define $\overline{m}$ as a term in the language of $PA$ using the following recurrence.
$$\overline{0} = 0$$
$$\overline{m + 1} = S(\overline{...
- 15.7k
5
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1
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Algebraic number theory in first-order arithmetic
I was inspired to think about how algebraic number theory could be developed in first-order arithmetic, since most developments of ANT do make use of complex numbers. Most of the time such uses of ...
- 26.8k
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Peano axioms: S(1)=1?
I was reading through the Peano axioms here, and a question came up:
Can we define $S(0)=1$, and $S(1)=1$?
It seems to me (at least as it is stated) that it would satisfy all of the axioms listed.
...
- 33
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What is Complete Induction without Hypothesis?: Ordering of $\mathbb{N}$ from Peano's Axioms
The following is from Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra; Edited by H. Behnke, F. Bachmann, K. Fladt, W. Suess and H. Kunle
I've used ...
- 3,349
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Is Fermat's last theorem provable in Peano arithmetic?
The sentence $S$ which Gödel in his proof of the incompleteness theorem proves to be be unprovable in the system of Peano arithmetic can be proved (as a true theorem of PA) outside PA (and necessarily ...
- 18.2k
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Peano Axioms for Arithmetic
I have to give a talk which mentions the Peano Axioms for Arithmetic. I want to make sure when i am explaining them in English I am describing them correctly.
(PA1) $(\forall x_1)( \neg s(x_1) = 0)$ $...
- 133
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1
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Connection between number theory and the Von Neumann construction of naturals
There are many unsolved conjectures and hypothesis in number theory. For example, the twin primes conjecture, Goldbach's conjecture, the Riemann hypothesis, infinitude of Mersenne's primes, and many ...
- 2,458
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Defining Primes in Non-standard Models of Peano Arithmetic
I was recently reading a post basically discussing an "intuition" that Goldbach's Conjecture may be a statement which is undecidable (the post does not specify which axiomatic system the statement is ...
- 3,270
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Stuck on GEB chapter 9 - is b a MU number? is b a TNT number?
I'm reading through Gödel, Escher, Bach, and I found myself stuck at chapter 9. I've been rereading through several times already, but I must be missing something. To clarify my background, I'm a ...
- 163
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Why don't we use Presburger's arithmetic instead of Peano's arithmetic?
I was reading about quantifier elimination and discovered the Presburger Arithmetic, the article mentions two points about it:
It is decidable, complete and consistent.
It omits multiplication ...
- 24.2k