All Questions
5
questions
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2
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306
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How to prove that there are $n$ natural numbers that are less or equal than $n$ and what properties are allowed to use in induction.
Let $n \in \mathbf{N}$. I wondered how to prove that there are exactly $n$ natural numbers that are smaller or equal than $n$. This seems somewhat circular which confuses me. I guess the way to do ...
- 389
1
vote
1
answer
55
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Finiteness, finite sets and representing its elements.
A set $S$ is called finite if there exists a bijection from $S$ to $\{1,...,n\}$ for one $n \in \mathbf{N}$. It is then common to write its elements as $s_1,...,s_n$. I now wonder, why this is ...
- 389
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votes
3
answers
258
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Can the natural numbers contain an element that is not representable by a number?
I read the following document: https://www.math.wustl.edu/~freiwald/310peanof.pdf . In this document, the author wants to formalize that natural numbers, that are informally thought of as a collection ...
- 389
1
vote
1
answer
216
views
How to show that a triple $(P, S, 1)$ constitutes a Peano System?
Mendelson (in Number Systems & the Foundations of Analysis) defines a Peano System as a triple $(P, S, 1)$ consisting of a set $P$, a distinguished element $1 \in P$, and a singulary operation $S :...
- 260
2
votes
1
answer
220
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Determine whether or not the following structure $(P,S,1)$ is a Peano System
First this is how the book define as a Peano System.
By a Peano System we mean a set $P$, a particular element $1$ in $P$, and a singulary operation $S$ on $P$ such that the following axioms are ...
