All Questions
27
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
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
1
vote
1
answer
76
views
Few short questions on notational choice
There are few questions on notational choice that seem to come up a number of times. It seems that, while some of this might be somewhat context dependent, it might be useful to get a general idea. I ...
- 1,022
1
vote
1
answer
141
views
Why does my book use $\subseteq$ instead of $\subset$ to describe a transitive set?
The following FOL statement is used in my book to describe a transitive set (which ultimately sets the stage for the definition of ordinals):
$z$ is a transitive set iff $\forall y \in z [ y \...
- 5,064
0
votes
1
answer
56
views
Confusion Regarding Notation in a Proof about two Well-Ordered Sets
This partial proof is taken from A Course in Mathematical Logic for Mathematicians by Yu. I. Manin.
Lemma. Let X and Y be two well-ordered sets. Then exactly one of the following alternatives holds:
(...
- 1,610
2
votes
0
answers
80
views
What does $\dot{-}$ means
I was reading Accesible Independence Results for Peano Arithmetic, Kirby & Paris and saw a symbol (at top of the $\mbox{page 288}$, in $\mbox{Lemma 3.}$) like $\dot{-}$ which I do not know what it ...
- 445
0
votes
0
answers
46
views
Constructing a set with order type $\omega\cdot n$ and related notation
Let $A_n=\{0,1,\ldots,n-1\}\subseteq\mathbb{N},n>0$, so that $|A_n|=n.$ Order on this set is the usual order $<$ on the naturals. For this example, let's use $A_2=\{0,1\}$, $\mathbb{N}$ and ...
- 37
4
votes
3
answers
1k
views
What are $\aleph_0$, $\omega$ and $\mathbb{N}$ and how are they related to each other?
I have seen these three symbols, $\aleph_0$, $\omega$ and $\mathbb{N}$, a lot in my reading (mostly in analysis, I have very limited experience in set theory). I have seen in various places they are ...
user765539
2
votes
1
answer
104
views
Omega Notation in Explanation of Chain Rule
In his book Multivariate Calculus and Geometry, Sean Dineen explains the chain rule as follows:
I understood well enough his point but, then he goes on to introduce an unknown and unexplained $\...
- 47
0
votes
2
answers
188
views
What is the meaning of $L_{\alpha}$ and $L_{\alpha}[x_1, x_2, \ldots, x_{n-1}, x_n]$ notations in relation to Infinite Time Turing Machines?
I thought that the $L_{\alpha}$ notation denotes a set of reals that can occur on the output tape at stage $\beta$, where $\beta$ is any ordinal less than $\alpha$ and the input is an arbitrary real. ...
- 922
8
votes
2
answers
393
views
Why are ordinals multiplied in reverse order
When two ordinals, $\alpha$ and $\beta$, are multiplied together, $\beta$ is taken as the most significant multiplicand and $\alpha$ as the least significant multiplicand in the product $\alpha \cdot \...
- 1,815
-2
votes
1
answer
45
views
Unknown notation/omega
What does the 3rd term in $\omega\times\omega\times\omega^{\operatorname*{\omega}\limits_{\smile}}$ with semicircle below the last $\omega$ in definitions 5.3.6
here mean?
- 3,978
0
votes
0
answers
66
views
Boolean algebras/Unknown notation
Does someone know what is meant (in the context of trees and Boolean algebras by Shelah) here on the page 8 right above Remark 1.5:
$$\{\langle\rangle\}\cup\{\langle\xi\rangle\otimes_{\zeta(*)}d\eta:\...
- 3,978
1
vote
1
answer
422
views
Is there a standard notation for an ordinal number with cardinality of the continuum?
Under ZFC, the real numbers can be well-ordered. So, there is some ordinal number whose cardinality is that of the continuum. Is there a standard notation for this number?
For example, the first ...
- 5,256
1
vote
1
answer
80
views
Function with an ordinal domain: an ambiguity between a notation for a value and for an image
Let $f\colon\alpha\to\beta$ be a function from an ordinal $\alpha$ into an ordinal $\beta$. Since ordinals are transitive sets (i.e. a set $x$ so that $\forall y(y \in x \Longrightarrow y \subseteq x))...
- 4,528