Attempt at a coherent question here. Maybe best to leave it simple. What is an infinite ordinal? — tim wood
0 1 2 3 4 5 ... ^ ^ ^ ^ ^ ^ ... | | | | | | ... v v v v v v ... 0 2 1 3 4 5 ...
The system Ω of all [ordinal] numbers is an inconsistent, absolutely infinite multiplicity. — Georg Cantor
I have to pause here because you have said explicitly what I thought was an error on a video.[/url]
What vid, I'll take a look. Youtube giveth and Youtube taketh away.
— tim wood
An ordering has a first element (yes?).[/url]
A well-ordering, yes. Orderings in general need not have first elements, for example the usual < on the integers. But well-orderings always have a first, second, third, ..., element.
— tim wood
At some point candidates for the second element are exhausted, how then going forward does it remain uniquely orderable? — tim wood
And if it does, then how do you get to ε? — tim wood
Btw, for clarity on a difficult subject, you're hitting it out of the park! — tim wood
OK, I'm not reading all that... but thank you. — Banno
↪fishfry Well done! Handy reference for an oldtimer, Too. :cool: — jgill
An ordering has a first element (yes?) — tim wood
At some point candidates for the second element are exhausted, — tim wood
uniquely orderable — tim wood
Hmm. New for me: well-ordered does not mean in-order, yes? — tim wood
! Is it correct to think of all the well-orderings to be the same thing as all the permutations? — tim wood
For the natural numbers, that would just be ω! (or maybe better aleph-0!), yes? The image I have - that maybe I have to work through - is of something like a deck of card. Fifty-two of them. With 52! possible arrangements. With the cards, at least, you can't get past 52! arrangements without duplication. — tim wood
I suppose similarly there is an upper limit on the arrangements of NN. And I get it that each of those is countable. Why not, they're just arrangements. — tim wood
Again how to be brief? — tim wood
I might not be the best person to give advice about that :-)
Following with your construction above, it seems that ω is the ordinal associated with NN. — tim wood
That leaves the question, how many infinite subsets of NN are there? — tim wood
(And with each ω as a subset, if well-ordering does not require being in-order, then each of those yielding ω! permutations - yes?) — tim wood
If all of this ends at ε - does it end at ε? - — tim wood
then how do you get beyond ε and still be countable? — tim wood
If it doesn't end, how do you get larger ordinals. — tim wood
I'm still reading. I suppose the answer is up ahead. I'll look for it, then, and wait for it. — tim wood
I was thinking naively that well-ordering means the set can be and is ordered lexicographically. — tim wood
But with the OP I'm thinking the "and is" isn't part of it. E.g., I thought 1,2,3 is well-ordered when presented as {1,2,3}. But now I'm thinking that 1,2,3 is well ordered in each, and all, of six variations. If the latter is true, then my "uniquely orderable" is just a mistake. — tim wood
First? Or least but not necessarily first? — tim wood
can be and is — tim wood
well-ordering means the set [...] is ordered lexicographically. — tim wood
1,2,3 is well ordered in each, and all, of six variations — tim wood
Every permutation of {1 2 3} "induces" a distinct well ordering of {1 2 3}. — TonesInDeepFreeze
Is it correct to think of all the well-orderings to be the same thing as all the permutations? — tim wood
ω! — tim wood
NN — tim wood
seems that ω is the ordinal associated with NN. But it also seems that ω is also associated with every infinite subset of NN. — tim wood
how many infinite subsets of NN are there? — tim wood
ε — tim wood
how do you get beyond ε and still be countable? — tim wood
Here is some of the terminology (not necessarily in logical order) that one must have a very clear understanding of in order to have a clear understanding the matters in this thread. — TonesInDeepFreeze
Seems like every popular leftist finite ordinal is coming out as trans these days. Pretty soon they’ll make it illegal to be a cis finite ordinal at all! — Pfhorrest
And one can get the "general idea" about ordinals — TonesInDeepFreeze
a clear understanding — TonesInDeepFreeze
The idea was that they were simply defined into existence, — tim wood
Question: is the change from ω-street to ε-street a "can't get theah from heah" transition? — tim wood
I see the language that says you just add a successor, but what successor would that be? — tim wood
Your OP a Disneyland of rides, and I not tall enough for most of them. — tim wood
And yet it's countable. That seems strange. — tim wood
With zero and 1, I take it a person can get to any number in {0, 1, 2,.., n}, though perhaps not efficiently. The limit of that being ω. Hmm. The only way I can understand ω or ω+1, is simply as the numbers ω and ω+1, which are just larger than any of the {0, 1,.., n}. — tim wood
Maybe I need a bit more care in thinking about what a number is. Transfinite cardinals and ordinals thus not numbers in any naive sense, but in an extended sense, perfectly useful and thus perfectly good. — tim wood
An ordinal number is the order-type of a well-ordered set — fishfry
A well-order is an order relation on a set such that every nonempty subset has a smallest element. — fishfry
Every possble nonempty set of the naturals has a smallest member, so the naturals are well-ordered by <. — fishfry
the negative numbers have no smallest element. — fishfry
ω stands for the natural numbers in their usual order. — fishfry
If the usual order 0, 1, 2, 3, ... is called ω — fishfry
our funny order (N,≺) is called ω+1 — fishfry
even without choice, some uncountable sets can be well-ordered. Just not all of them. — fishfry
in the absence of choice, there are infinite sets that are not well-ordered — fishfry
there's also no set of all sets bijectively equivalent to the naturals or any other cardinality. — fishfry
the Alephs are indexed by whole numbers — fishfry
Infinity and the Mind by Rudy Rucker. — fishfry
in the limit, or sup operation. — fishfry
Every permutation of a finite set is a well-order. — fishfry
A well-order is an order in which every nonempty subset has a smallest element. — fishfry
So for infinite sets, permuting does not preserve order type. — fishfry
there are many who consider such books "archives" and that the sine qua non of learning is the good teacher. I myself favor middle ground, finding books to allow triangulation on a topic by one providing illumination where another is dark, while a teacher provides guidance and explains difficulties. — tim wood
I vaguely remember a comment that "explained" certain large cardinals by saying, "You can't get theah from heah." By which I understood that no ordinary process could get to them, meaning, as best I get it, that no recursion scheme could get to them. — tim wood
The only way I can understand ω or ω+1, is simply as the numbers ω and ω+1, which are just larger than any of the {0, 1,.., n}. — tim wood
Maybe I need a bit more care in thinking about what a number is. — tim wood
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