All Questions
10
questions
14
votes
8
answers
2k
views
Why can't we define arbitrarily large sets yet after defining these axioms? (Tao's Analysis I)
In Tao's Analysis I I am very confused why he says we do not have the rigor to define arbitrarily large sets after defining the below 2 axioms:
Axiom 3.4 If $a$ is an object, then there exists a set
$...
- 3,019
0
votes
2
answers
306
views
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
0
votes
0
answers
103
views
Why is the principle of induction for natural numbers not "self-evident"? [duplicate]
The principle of induction can be stated, in first-order logic, as follows. Let $S\subseteq\mathbb N$, and suppose that
$0\in S$.
$\forall n:n\in S\to n+1\in S$.
Then, $S=\mathbb N$. Now, suppose ...
- 20.7k
3
votes
3
answers
187
views
Infinite natural numbers?
Only using the successor function $\nu$ and the other axioms, how do we guarantee that the "next" generated number is different from all the "previous" numbers (I am using ...
- 325
0
votes
0
answers
118
views
A Doubt about Well Ordering Principle and Principle of Mathematical Induction
I have had this lingering doubt in my mind for a very long time: One of the standard constructions of N starts by assuming the 5 Peano Axioms, proving that every non-zero is a successor and s(n) is ...
- 21
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
702
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"Proof" of $0=1$ in set theory [closed]
Ok, so here is a proof of "$0 = 1$" I came up with today. You can do in set-theory, where natural numbers are defined in the usual way.
Proof: Let $\mathsf{Succ}$ be the function that takes ...
- 3,123
3
votes
2
answers
1k
views
Finite Ordinals and Natural Numbers
I'm studying set theory and I'm focusing on von neumann ordinals. I've built an understanding of the reasoning that brings to the set-theoretic construction of the natural numbers whose soundness I'm ...
- 2,611
2
votes
1
answer
529
views
Why is the definition of inductive set well defined?
I've been studying from Enderton's Mathematical Introduction to Logic in which he defines an inductive set as follows:
To simplify our discussion, we will consider an initial set $B \subseteq U$ ...
- 303
4
votes
3
answers
3k
views
Is there a natural number between $0$ and $1$?
Is there a natural number between $0$ and $1$?
A proof, s'il vous plaît, not your personal opinion. (Assume the Peano Postulates.)
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