All Questions
25
questions
2
votes
2
answers
91
views
If I have a sequence $a_0, a_1, a_2, \cdots$ , then is expressing the limit of this sequence as $a_\omega$ sensible?
If I have a sequence created by some rule which comes to a limit , then I can express it as $a_0, a_1,a_2,\cdots$.
If I said $\lim_{n \to \infty} a_n = a_{\omega} $ , is that a sensible thing to do ?
...
- 4,655
2
votes
1
answer
156
views
Is this a valid basis for a transfinite number system?
I've been curious about transfinite number systems including infinite ordinals, hyperreals, and surreal numbers. The hyperreals in particular seem particularly appealing for introducing a hierarchy of ...
1
vote
1
answer
116
views
$\omega$-th or $(\omega + 1)$-th when putting odd numbers after even numbers?
The following clip is taken from Chapter 4 - Cantor: Detour through Infinity (Davis, 2018, p. 56)[2]. When putting odd numbers after even numbers, what should the index for the first odd number ($1$ ...
- 103
2
votes
1
answer
105
views
Taking the limit beyond infinity, with the ordinals
Imagine a function $f:X\to X$ and $x\in X$ (keeping $f$ and $X$ vague on purpose) and let's define
$u_1 = f (x)$
$u_2=f^2(x)=f(f(x))$
$u_n=f^n(x)$
Let's also assume that $\forall n, u_{n-1} \neq u_n $ ...
- 169
3
votes
1
answer
136
views
Why is $\epsilon_0$ a fixed point? Why don't/can't we define tetration of $\omega$ beyond $\omega$?
I'm trying to learn about transfinite ordinals and got stuck here.
Once you've added $1$ infinitely, you can add another $1$ and get a larger result. If you take the successor of $$1+1+1+1+\dots = \...
- 157
0
votes
1
answer
93
views
Help understanding countable and uncountable infinities
just had some questions about countable and uncountable infinities.
If we take a limit that results in $\frac{ \infty }{0}$, we typically conclude that the limit is just $\infty$, correct? But if the ...
- 515
0
votes
0
answers
83
views
Does the universal set operate as the Von Neumann universe in non well founded set theories?
I've been researching non well founded set theories (E.g. NF, NFU, etc.) and have been wondering if there are any similarities between the universal set & Von Neumann universe ? Or if there the ...
- 19
1
vote
0
answers
39
views
Can the existence of $\omega$ be derived from this alternate axiom of infinity? [duplicate]
Suppose we use the following version of the axiom of infinity in ZFC:
$$\exists x:(\varnothing\in x)\wedge(\forall y\in x:\exists z\in x:y\in z).$$
In words, there is set containing the empty set, in ...
- 6,694
0
votes
1
answer
115
views
How can ordinal arithmetic violate the standard approach to arithmetic with infinites?
From what (little) I understand of this Wikipedia article, where $\omega$ denotes the ordinal "identified with" $\mathbb{N}$, and $\aleph_0$ is the cardinality of $\mathbb{N}$, and $\...
- 42.7k
1
vote
2
answers
90
views
Is there any use to $\mathbb{N}$ as a sort of ring mod an infinite number?
I was recently joking with my younger brother regarding rings as subsets of $\mathbb{N}$, and the use of such a construction (don't laugh- we have a weird sense of humor). The discussion got me ...
- 1,789
0
votes
2
answers
209
views
What is the rank of the set of functions from $\omega$ to $\omega$?
My stab was to say each member is a set of pairwise distinct ordered pairs of $\omega$ into $\omega$, so each function has rank $\omega + 1$ , therefore the entire set of functions has rank $\omega+ 2$...
- 149
1
vote
1
answer
403
views
Please clarify the pun in an equation with omega and infinity [closed]
I received the following equation:
$\Large \sqrt[\Omega]{\omega} = \infty$
This shall be some kind of mathematical joke but my math (or physics) is not advanced enough to see it. Can someone help?
- 113
-3
votes
2
answers
102
views
Why is $\varepsilon_1$ not smaller? [closed]
I'm thinking that $\varepsilon_1$ could in theory be $lim[\omega+1,\omega^{\omega+1},\omega^{\omega^{\omega+1}},...]. $
This, in my opinion, would be smaller than the normal definition of $\...
1
vote
1
answer
134
views
Cardinality of powers of infinite sets
How to prove that for infinite sets $S,S'$-
(i) If $|S'|<|S|$, then $|S^{S'}|=|S|$?
(ii) If $|S'|=|S|$, then $|S^{S'}|=|2^S|$?
Also what is $|S^{S'}|$ when $|S'|>|S|$?
- 165
0
votes
1
answer
68
views
Finding $\omega$ with an arithmetic series
I just watched a really interesting video by Vsauce about counting past infinity here, and from which I have now learned that the next number after infinity, is $\omega$, then $\omega+1$, and so on.
...
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