And still curious whether you understand now that "aleph_1/(2^aleph_1)" is nonsense. — TonesInDeepFreeze

Yes, I got that. Thank you very much for your explanation.

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

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.

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.

What I have in mind is that I simply divide the interval by 2^infinity in one step — MoK

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.

Imagine I define a number line as comprising only the numbers - integers - 0, 1, 2,.... You want to divide the interval between, say, 0 and 1, using 1/2, or 2/3 as examples of what you have in mind. Two possible results: 1) you have demonstrated that my number line is impoverished by showing fractions, or 2) I simply say, not on my number line.

Now the number line made up of reals, with uncountably many points between 0 and 1. You would like to divide that up so that there is no point between two points sufficiently close together.so that there is no point between them, thus zero "real" distance between them. It's easy enough to specify an example of one point of that pair, call it .75. What's the (name of the) next point so close no other point is between the two?

In the case of my number line of integers only, I might say not on my number line, and you might argue that rational numbers are clearly members of my line, as a consequence of being representations of simple ratios of the numbers themselves. Thus my claim defective on its face.

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.

Is f(infinity) a member of the above sequence? — MoK

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.)

how could the sequence be an infinite one? — MoK

How can you ask that question when I defined the domain of f to be the set of positive natural numbers? (The set of positive natural numbers is infinite, and if the domain of a relation is infinite then the relation is infinite.)

You really need to learn basic mathematics.

(Also: A sequence is a certain kind of function, and a function is a certain kind of relation, and a relation is set of ordered pairs. So, f(n) is not a member of the sequence f; rather <n f(n)> is a member of the sequence. f(n) is an entry in the sequence (i.e. f(n) is in the range of the sequence).

And still curious whether you understand now that "aleph_1/(2^aleph_1)" is nonsense.
— TonesInDeepFreeze
Yes — MoK

Actually, No:

I simply divide the interval by 2^infinity — MoK

(1) What do you mean by "divide the interval"? At least four options:

(a) Partition the interval

(b) Divide the cardinality of the interval

(c) Divide the distance of the interval

(d) Deploy an operation that divides, then divides the result ... infinitely many times, as a single combined operation that has a final value

Option (d) is meaningless. You are trying to argue against the proof that for every positive real number x (= the distance of the interval [0 x]) there is a positive real number y such that 0 < y < x. Your confused, ill-premised, ignorant and incoherent idea is that by dividing the interval 2^aleph_1 number of times, we get an i such that for any real number y>0, we have 0 < i < y.

But you have not defined such an operation. All you've done now is to claim you understand my explanation of cardinal division while going right past it to repeat your same mistake, but with even greater vagueness by insisting on an undefined "infinity" as a divisor.

You're not only not making progress, but you're going in the opposite direction of progress.

You really really need to learn basic mathematics.

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

Not a question of "validity". It's just that it is not defined by you. Your method is to throw around mathematical terminology without understanding it, thus to combine it in ways that are not coherent.

You really really need to learn some logic, set theory and the mathematics of standard analysis and non-standard analysis.

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

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

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

As you touch on, the system of naturals can be extended to a system of rationals, and the system of rationals can be extended to the system of reals. And each is a different system. And the system of reals can be extended to a system with infinitesimals. And that is a different system. But when we extend in that way, we do it coherently. We prove the existence of the systems - the sets of numbers and the key operations and relations and the key properties of those). But @MoK's own offering is not coherent - does not proceed by coherent definitions and proofs. He needs to learn at least the basics of the real number system, then he can look up 'non-classical analysis' to see how an actually coherent and rigorous development of infinitesimals is done.

An ordinary route:

Prove the existence of a system of naturals, and define a particular system of naturals.

Prove the existence of a system of integers, as there is a subsystem isomorphic with the system of naturals, and define a particular system of integers.

Prove the existence of a system of rationals, as there is a subsystem isomorphic with the system of integers, and define a particular system of integers.

Prove the existence of a system of reals, as there is a subsystem isomorphic with the system of rationals, and define a particular system of reals.

Prove the existence of a system of hyperreals, as there is a subsystem isomorphic with the system of reals, but we cannot define a particular system of hyperreals (at least not with the method of compactness or the method of ultrafilters).

It seems that there is no operation of infinite division in the real number system. That was something I didn't know. — MoK

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.

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

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.

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

The continuum is a mathematical abstraction. It has no representation or instantiation (as far as we know) to anything in the physical world.

This is all explained by the mathematical theory of limits. The sequence 1/2, 3/4, 7/8, 15/16, ... is an infinite sequence that has the limit 1. The "top of the cake" is the limit of the sequence. In terms of a cylinder, the very top would be a circular disk of zero thickness, the same as any other horizontal slice. That is, the intersection of a cylinder with a horizontal plane (parallel to the top and bottom of the cylinder) is a circular disk.

It can be confusing to think about cakes, because cakes are made of atoms; and the Planck limits preclude our making fine enough horizontal slices to produce a zero-thickness slice. Cakes are not cylinders, and Math $\neq$ Physics!

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

There is no evidence that anything in the physical universe is a mathematical continuum. It's possible that there is, but this is a deep open question that (with respect to current science) is more philosophical than scientific.

How imaginary (perhaps you mean complex) numbers related to geometry is a pretty cool subject, but far afield from understanding the nature of limits.

Here's the coolest example I know of how complex numbers relate to the geometry of the plane.

Take a regular n-gon (triangle, square, pentagon, hexagon, heptagon, octagon, etc.)

Place it in the complex plane such that its center corresponds to the origin of the plane, the complex number 0; with one of the vertices as the point (1,0) in the plane, or the complex number 1.

Now the vertices of the n-gon are exactly all complex n-th roots of 1; that is, they are all the complex solutions to the equation $z^n = 1$.

As a concrete example, consider a regular 4-gon, or square, with one vertex at the point 1 in the complex plane. Where are the other vertices? At $i$, $-1$, and $-i$. And these are exactly the four complex numbers whose fourth power is 1.

And this works for any regular n-gon. The vertices of the regular 17-gon are the seventeen 17th complex roots of 1.

That's one of the coolest things I know. And it is one example of the deep relation between complex numbers and geometry, which is the question you asked. But it's got nothing to do with the definition of limits or the nature of the mathematical continuum, but it's definitely interesting. I only mentioned it since you asked about the relationship of complex numbers to geometry.

Here's some Wikitude on the subject.

https://en.wikipedia.org/wiki/Root_of_unity

None of this has ANYTHING to do with relativity or spacetime or the physical world. Physicists use the mathematical continuum to model spacetime, but there is no evidence whatsoever that time or space or spacetime are literally the same as a mathematical continuum. They use the mathematical continuum as an approximation that seems to work, to the limit of our ability to measure the results of our experiments.

Remember: Math $\neq$ Physics!

Thanks for the superb reply. The reason i brought up Hawking's "no boundary" thesis is that i was thinking maybe geometry and limits are incomplete by themselves and need the 4 spatial dimensions and 1 time dimension in order to make sense of it. That is, mathematicians assume math can stand on its own, but maybe it can't. However i also now see how the physics can be in trouble where maybe the math isn't. You've explained with the cylinder how the top of it is the limit such that if i metaphorically touch it with my finger i am touching a point limit. However if we bring in time and do the series Zenonian as i proposed (one at a time), and with each new slisce changed the color of the new slice, i can ask "what color" the top of the cylander would be. This causes a problem *because* it is a process and processes aren't used like that in mathematics. But again, Hawking had the thesis from the 80's that 5 dimensions (4 spacial Euclidean ones and 1 temporal one that acts as space and uses "imaginary time" as he says) wherein there is no before of time (as there is no north of the North Pole) express a hologram such that 2 dimensions are projected from the 5 dimensions infinitely far away. I know it's unorthodox, but why can't this been seen purely from it's mathematical side and brought into mathematics itself? Hawking explained away indeterminacy with this idea.The lines here seem rather blurry to me, but i read of mathematics mostly from the historical perspective, although i started working through a discrete mathematics textboom recently.

I will be thinking about your reply and other posts on this thread throughout the day

Thank you very much for your contribution to my thread. I learned lots of things. You saved me a lot of time as well.

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

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.

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

Just out of curiosity, why aleph_1 is uncountable?

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

I agree with what you stated.

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

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. :)

By infinity, I mean aleph_0. — MoK

Then say 'aleph_0' since, in this context, 'infinity' is ambiguous.

I thought that the value of f(alep_0)=0 — MoK

You think that only if you didn't read my post.

For the third time: The domain of f is the set of positive natural numbers, therefore, aleph_0 is not in the domain of f, therefore "f(aleph_0)" is meaningless.

one could define a sequence g(n+1)=g(n)/10 where g(0)=1 — MoK

What is the domain of g? If the domain of g is the set of natural numbers then:

g(0) = 1
for all natural numbers n, g(n+1) = g(n)/10

So g, just like f, converges to 0.

But g(aleph_0)=0.0...1 — MoK

So the domain of g is not the set of natural numbers. What is the domain of g?

I explained to you before that ".0...1" represents a sequence h on w+1 such that:

For all natural numbers n, h(n) = 0
h(w) = 1

So if aleph_0 is a member of the domain of g, then:

g(0) = 1
for all natural numbers n, g(n+1) = g(n)/10
g(aleph_0) = h

But we don't have a less than relation defined that includes h in the field of the relation, so no definition of "converges".

And are you going to be defining the domain and range of g to include other objects?

and we find g(aleph_0) >0 — MoK

You haven't defined ">" so that it includes g(aleph_0) in the field of the relation.

so f(aleph_0)>0 as well. — MoK

Nonsense. aleph_0 is not in the domain of f. "f(aleph_0)" is meaningless notation.

I am sure you can define things better and provide a better argument. — MoK

Only because my meager knowledge at least includes an understanding of what such concepts as a function, domain, range, value of a function at an argument, sequence, infinite sequence, alephs, etc. mean. Meanwhile, your knowledge about these plainly required things is less than meager.

You need to learn at least the minimal basics of this subject. I mean, you don't even know what "domain of a function" means.

why aleph_1 is uncountable? — MoK

That you ask that question, indicates that you don't know what 'aleph' and 'countable mean' despite that you toss those terms around as if you're making actually meaningful statements with them.

Df. S is countable if and only if (S is one-to-one with a natural number or S is one-to-one with w)

Df. S is uncountable if and only if S is not countable

Df. T strictly dominates S if and only if (there is an injection from S into T & S is not one-to-one with T).

Notation: If j is an ordinal, then j+ is ju{j}.

Definition (by transfinite recursion on the ordinals):
aleph_0 = w
aleph_j+ = the least ordinal that strictly dominates aleph_j
If j is a limit ordinal, then aleph_j = U{alelph_m | m e j}

1 = 0+, so aleph_1 = the least ordinal that strictly dominates w. Since aleph_1 strictly dominates w, aleph_1 is not one-to-one with a natural number nor is S one-to-one with w. So, aleph_1 is uncountable.

/

Please get a book on the first order predicate calculus, then one on set theory.

It exists in your mind, your imagination, but not in the physical world — T Clark

Does that mean the mind is also an abstraction? Something outside the physical world? If so how does one explain what happens to my mind when you crush my head between two boulders?