All Questions
Tagged with peano-axioms elementary-number-theory
50
questions
0
votes
2
answers
143
views
Help me check my proof of the cancellation law for natural numbers (without trichotomy)
can you guys help me check the fleshed out logic of 'my' proof of the cancellation law for the natural numbers? It's in Peano's system of the natural numbers with the recursive definitions of addition ...
- 171
0
votes
1
answer
160
views
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
-1
votes
2
answers
110
views
Applying Peano Axioms to Subsets of Natural Numbers [duplicate]
Concise Question
Is it possible to apply the Peano axioms to subsets of the natural numbers and thus define arithmetic operations on those subsets strictly in terms of those subsets? If so, is it also ...
- 123
1
vote
0
answers
78
views
Question regarding natural numbers in Tao’s Analysis 1.
This doubt might be a mistake on my part, but its confused me for a while now so I decided to ask here.
In Terrence Tao’s “Analysis 1”, at the beginning of the chapter about Natural Numbers (where he ...
- 89
1
vote
0
answers
101
views
Check my work on the the following theorem is correct
Prove that if $m \leqslant n$ then $\exists!\ p\in\omega$. such that $m+p=n$
Correction:induction on $n$ thanks to @Brian M. Scott
Working assumptions:
6.1 Definition By the set of the natural numbers ...
user837396
1
vote
1
answer
66
views
Proving that $\mathbb{Z_{-}} \cap \mathbb{N}=\emptyset$
In assumption that $\mathbb{N}$ and successor function ($\overline{x}$) over $\mathbb{N}$ is defined by 5 Peano axioms:
$1\in\mathbb{N}$
$n\in\mathbb{N} \Rightarrow \overline{n}\in\mathbb{N}$
$\...
- 13
0
votes
1
answer
117
views
Peano arithmetic - Why is adding n to m the same as incrementing m n times?
Addition(+) is defined using the successor function(++) in Peano arithmetic as:
0 + m = m
(n++) + m = (n + m)++
While these are intuitive axioms that are consistent with my previous, elementary, ...
user487586
2
votes
1
answer
117
views
How to prove $x=5$ using the Peano Axioms?
This question relates to Proposition V of Gödel's 1931 Incompleteness theorem (and another one posted on math.stackexchange here ) which says:
For every recursive relation $ R(x_{1},...,x_{n})$ there ...
- 571
2
votes
4
answers
490
views
How does the Peano axiom of induction prevent S-loops?
First, let me state what I understand to be the first-order rendition of Peano's 5th axiom: the axiom of induction.
For all natural numbers, for any relation/property/predicate $R$...
$$(R(0) \land \...
- 571
0
votes
1
answer
58
views
Can this function be described by a formula?
Suppose $PA$ is Peano arithmetic. For $m \in \mathbb{N}$ define $\overline{m}$ as a term in the language of $PA$ using the following recurrence.
$$\overline{0} = 0$$
$$\overline{m + 1} = S(\overline{...
- 15.7k
0
votes
1
answer
83
views
Is Peano's axiom of induction needed to show $n^\prime\ne n^{\prime\prime}$?
This is the statement of Peano's axioms I will assume for this discussion:
$1$ is a number.
To every number $n$ there corresponds exactly one number $n^\prime.$
$n^\prime=m^\prime\implies n=m.$
$n^\...
- 3,349
8
votes
3
answers
434
views
Are there "interesting" theorems in Peano arithmetic, that only use the addition operation?
More precisely, are there "interesting" theorems of Presburger arithmetic, other than the following four well-known "interesting" ones?
The commutativity of addition.
The theorem stating there are ...
- 109
2
votes
1
answer
262
views
In the extension to integers, how should this restriction on the domain of natural numbers be interpreted?
My interpretation of the sub-paragraph in the book which I am asking about is this:
BEGIN SUMMARY
Note that in this context 0 is not considered to be a natural number.
The pairs of natural numbers $...
- 3,349
1
vote
0
answers
82
views
How is it shown using the additive cancellation law that $x\mapsto x+h$ has an inverse?
See Fundamentals of Mathematics, Volume 1 page 101, for context.
This is one of those facts that is so obvious that I find it difficult to prove. My question regards part of the proof of the ...
- 3,349
2
votes
1
answer
48
views
Is this a valid proof that $\forall_{x,y}\left[x\ne y\implies x<y\lor x>y\right]$ in $\mathbb{N}_1$?
I do not consider the following to prove all of the so-called trichotomy of order, since I take that to be a statement that exactly one of $<.=,>$ holds for any given pair of numbers.
The ...
- 3,349