Yes, I got that. Thank you very much for your explanation.And still curious whether you understand now that "aleph_1/(2^aleph_1)" is nonsense. — TonesInDeepFreeze
That is a correct interpretation if you divide the interval by two, then another time divide the result by two, ad infinitum. What I have in mind is that I simply divide the interval by 2^infinity in one step. This operation seems to be invalid though to mathematicians.That's what I understand as a limit (process) - that can be approached but never reached, but if it could be reached, would yield whatever - the process of whatevering being nothing in itself, but useful as a tool. — tim wood
It is better to say greater than any finite number given the definition of a countable set in mathematics.Greater than any countable number or greater than any finite number? — TonesInDeepFreeze
It seems to me - subject to correction - that you cannot even reasonably think about that without at least giving a somewhat rigorous definition of what you think a - your - number line is, in the sense of what comprises it, or what it's composed of or made of. If points, then you have to decide how many, and at the least you run into a labeling problem if you have too many.What I have in mind is that I simply divide the interval by 2^infinity in one step — MoK
Is f(infinity) a member of the above sequence? — MoK
how could the sequence be an infinite one? — MoK
And still curious whether you understand now that "aleph_1/(2^aleph_1)" is nonsense.
— TonesInDeepFreeze
Yes — MoK
I simply divide the interval by 2^infinity — MoK
What I have in mind is that I simply divide the interval by 2^infinity in one step. This operation seems to be invalid though to mathematicians. — MoK
Greater than any countable number or greater than any finite number?
— TonesInDeepFreeze
It is better to say greater than any finite number given the definition of a countable set in mathematics. — MoK
With the real number line, it seems to me it is the line itself that says you cannot. And if you say you can, then it is up to you to show how. — tim wood
It seems that there is no operation of infinite division in the real number system. That was something I didn't know. — MoK
Oh yeah, I can guess that. We, physicists, work with the infinities all the time. Of course, mathematicians do not agree with how we deal with infinities but strangely physics works. :) — MoK
If i cut a cake horizontally starting from the halfway point upwards with each slice being half the size of the one immediately below, what would the top of the cake look like? Isn't it indefinite? But you can definitely look at the cake, from all angles, and see that it has definite position in relation to its parts. So how do we reconcile the indefinite with the definite? I think this is what must be asked about the continuum. — Gregory
Hawking would say that four dimensional Euclidean space, with a time dimension that both 1) acts as space, and 2) is described by imaginary numbers, gives an answer to this question. That is to say, the universe as a whole gives the answer to the continuum. But how do imaginary numbers relate to geometry? — Gregory
By infinity, I mean aleph_0. I thought that the value of f(alep_0)=0 which is why I asked for its value. But after some thinking, I realize that it is not. In fact, one could define a sequence g(n+1)=g(n)/10 where g(0)=1. It is easy to see that for any value of n g(n)<f(n) except n=0 if (f(0)=1. But g(aleph_0)=0.0...1 and we find g(aleph_0) >0 so f(aleph_0)>0 as well. I am sure you can define things better and provide a better argument.What do you mean by "infinity" used as a noun?
There is the adjective "is infinite": S is infinite if and only if S is not finite.
And there are various infinite sets, such as:
the least infinite ordinal = {n | n is a natural number} = w = aleph_0
the least infinite cardinal = aleph_0 = w = {n | n is a natural number}
the least infinite cardinal greater than aleph_0 = aleph_1
card(set of functions from w into 2) = 2^aleph_0 = card(the power set of aleph_0)
card(set of functions from aleph_1 into 2) = 2^aleph_1 = card(the power set of aleph_1)
There are various uses of "infinity" as a noun, including such things as infinity as a point in the extended reals or infinity as a hyperreal in nonstandard analysis. But you've not specified.
In any case, how can you ask your question when I explicitly defined the domain of f to be the set of positive natural numbers? (Whatever you mean by "infinity" used as a noun, it is not a member of the set of positive natural numbers.) — TonesInDeepFreeze
Just out of curiosity, why aleph_1 is uncountable?Then just say "finite". But your particulars involve both countable and uncountable cardinals, all of which are greater than any finite cardinal but some of which are greater than any countable cardinal:
aleph_0 is countable
aleph_1 is uncountable
2^(aleph_1) is uncountable and greater than aleph_1 — TonesInDeepFreeze
I agree with what you stated.There are limits. As an example, consider the sequence 1/2, 1/4, 1/8, 1/16, ...
We all know that the limit of this sequence is 0. You can certainly call this infinite division if you like, as long as you understand what limits are. There are indeed infinitely many elements of the sequence, and you CAN think of this as "infinite division."
You can even think of doing it "all at once" if you like
What it means formally is that the elements of the sequence get (and stay) arbitrarily close to 0.
What it does NOT mean is that there is some kind of magic number that the sequence attains that is a "distance of 0" from 0, but is not 0. That's a faulty intuition.
In fact the formal theory of limits, once one learns it, is the antidote to all our non-rigorous, faulty intuitions about infinite processes.
Does that help, or perhaps refresh your memory? I'm pretty sure that physicists must be exposed to the formal theory of limits at some point. — fishfry
Yes, we are on the same page and thank you very much for your contribution. I learned a lot of things and refreshed my memory. :)Yes I understand and agree. But I am a little surprised that you seem to think that a sequence attains some kind of mysterious conclusion that lies at a distance of 0 from its limit, but is distinct from the limit. That's just not right.
Did my mention of limits ring a bell at all? Or raise any issues that we could clarify or focus on?
Because your idea of endless division is perfectly correct. But all that shows is that endlessly halving leads you to the limit of a sequence. But there's no "extra point" in there that's distinct from but at a distance of 0 from the limit.
Let me know if we're on the same page about this. — fishfry
By infinity, I mean aleph_0. — MoK
I thought that the value of f(alep_0)=0 — MoK
one could define a sequence g(n+1)=g(n)/10 where g(0)=1 — MoK
But g(aleph_0)=0.0...1 — MoK
and we find g(aleph_0) >0 — MoK
so f(aleph_0)>0 as well. — MoK
I am sure you can define things better and provide a better argument. — MoK
why aleph_1 is uncountable? — MoK
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