All Questions
6
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
4
votes
0
answers
115
views
Show that $f(a,b,c)=(a+b+c)^3+(a+b)^2+a$ is injective.
For a function $f : \mathbb{N} \times \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ defined by
$f(a,b,c)=(a+b+c)^3+(a+b)^2+a$
I want to show that $f$ is injective.
How can I show this?
I ...
- 41
0
votes
2
answers
78
views
Is this an adequate proof that any non-empty subset of N has a minimal element?
I am trying to improve my own standards for proof writing, but I cannot attend school, so I do not have the luxury of being able to speak to professors or peers to verify my attempts. In the proof ...
- 11
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
1
answer
79
views
Set of all finite subsets of $\mathbb{N}$ not equal to the to set of subsets of $\mathbb{N}$
I can kind of grasp why this is the case as if we take the union of all finite subsets of cardinality $i$ as $i$ runs through every natural number, we are listing finitely many elements each time.
...
- 459
0
votes
2
answers
592
views
Proof of the well-ordering principle
I tried to prove Well-Ordering Principle by myself, and I finally did it. However, I'm not sure if this proof is correct. Can anyone evaluate my proof?
Proof:
Since the set of natural numbers, $\...
- 123