can something exist literally 'infinitely' far away?

Originally posted by humy
∀x ∈ ℝ{≠L}

Should be
∀x ∈ ℝ{f(x)≠L}
as for whether it works, I need to give it some thought, but I think it does. I still don't like the 2 and am trying to think of a way to eliminate it.

Originally posted by humy
perhaps a better question is does either
2*∞ = ∞ is true
or
2*∞ > ∞ is false
make total sense in formal logic?

As before, it depends on your definition of ∞.

Originally posted by twhitehead
Should be
∀x ∈ ℝ{f(x)≠L}

Arr yes. That's what I really meant. So that should be;

lim {x→∞} f(x) = L ∧ L ∈ ℝ ∧ ∃x ∈ ℝ : f(x) ≠ L ∧ f(x) ∈ ℝ
⇒
∀x ∈ ℝ{f(x)≠L} : ∃y ∈ ℝ : y>x ∧ ∀z ∈ ℝ{>y} : 2*|f(z) − L| < |f(x) − L|

Actually, not sure but think it is slightly better to write that as:


lim {x→∞} f(x) = L ∧ L ∈ ℝ ∧ ∃x ∈ ℝ : f(x) ≠ L ∧ f(x) ∈ ℝ
⇒
∀x ∈ ℝ : f(x) ≠ L ⇒ ∃y ∈ ℝ : y>x ∧ ∀z ∈ ℝ{>y} : 2*|f(z) − L| < |f(x) − L|


Or, if you really desperately want to get rid of the arbitrary "2*", just a little less arbitrary I think would be
to change " ∀x ∈ ℝ : " to " ∀x ∈ ℝ{>1} : " and then just use " x* " instead of " 2* " thus:


lim {x→∞} f(x) = L ∧ L ∈ ℝ ∧ ∃x ∈ ℝ : f(x) ≠ L ∧ f(x) ∈ ℝ
⇒
∀x ∈ ℝ{>1} : f(x) ≠ L ⇒ ∃y ∈ ℝ : y>x ∧ ∀z ∈ ℝ{>y} : x*|f(z) − L| < |f(x) − L|

Originally posted by humy
I know the standard theory is that the universe is finite in size but unbounded but, just suppose you were told from a reliable source that the universe is infinite and unbounded and space is infinite in all directions.

Now, does it make any sense to say there might exist, say, a planet, that is literally an infinite distance from your current location? ...

I noticed the thread a little while ago and was thinking about this point. I can provide good justification for the answer yes and I can provide good justification for the answer no. Empirically, there is no way of justifying either answer. It's like asking what is on the other side of an event horizon - we can only make statements based on our theories. General Relativity allows space-times with both closed and open global topologies. So, either answer is possible. If we add the current lambda-CDM model and recent observation we have an expanding universe and the topology should be open. If lambda-CDM correctly describes nature, or at least describes nature well enough, then the universe has infinite extent and this implies that there can be objects arbitrarily distant from one another.

So our best theory seems to answer yes to your question, however one could argue that although there is no maximum distance objects can be from one another, if one chooses any two objects the distance between them will be finite. At this point we're into the realms of mathematical philosophy.

Basically, we have no way of knowing, we can in principle experimentally rule out theories that allow open topologies, however, current evidence is that the universe is expanding which implies infinite spatial extent. So we cannot use a theory to rule out the possibility. In special relativity we have the notion of causally connected. Two objects can be a finite distance apart, but with space expanding light can never get between them (we can see regions of space which must be causally disconnected from one another). So certainly in that theory there are objects between which light has an infinite travel time.

Originally posted by DeepThought
I noticed the thread a little while ago and was thinking about this point. I can provide good justification for the answer yes and I can provide good justification for the answer no. Empirically, there is no way of justifying either answer. It's like asking what is on the other side of an event horizon - we can only make statements based on our theori ...[text shortened]... So certainly in that theory there are objects between which light has an infinite travel time.

interesting. And I am kind of glad I am not the only scientifically minded person here not completely sure of the answer.

Originally posted by DeepThought
At this point we're into the realms of mathematical philosophy.

What do you mean by 'mathematical philosophy'? I see it as mathematical fact.

Basically, we have no way of knowing, we can in principle experimentally rule out theories that allow open topologies, however, current evidence is that the universe is expanding which implies infinite spatial extent.
How does an expanding universe imply infinite spatial extent? Most physicists actually seem to assume it is finite, so it seems unlikely that the implication is obvious to them.

If I am not mistaken, I have come up with a much better formula and this one works for defining the limit L, as x tends to infinity, for any function providing L is real thus finite;

lim {x→∞} f(x) = L ∧ L ∈ ℝ
⇒
∀x ∈ ℝ{>0} :
( ∃j ∈ ℝ : |f(j) − L| ≤ x )
∧
( ∃y ∈ ℝ : ∀z ∈ ℝ{>y} : |f(z) − L| ≤ |f(x) − L| )

Originally posted by humy
If I am not mistaken, I have come up with a much better formula and this one works for defining the limit L, as x tends to infinity, for any function providing L is real thus finite;

lim {x→∞} f(x) = L ∧ L ∈ ℝ
⇒
∀x ∈ ℝ{>0} :
( ∃j ∈ ℝ : |f(j) − L| ≤ x )
∧
( ∃y ∈ ℝ : ∀z ∈ ℝ{>y} : |f(z) − L| ≤ |f(x) − L| )

I believe I have just worked out how to make that even better"!

Even though the " |f(j) − L| ≤ x " and " ∀z ∈ ℝ>y : |f(z) − L| ≤ |f(x) − L| " parts have two completely unrelated purposes, while maintaining the validity of the formula, that formula can be changed to slightly modifiy its exact meaning to make it have a more eccentric meaning but one that allows it to be compressed down a bit by making "∃y ∈ ℝ " also do the job of "∃j ∈ ℝ " with:

lim {x→∞} f(x) = L ∧ L ∈ ℝ
⇒
∀x ∈ ℝ{>0} : ∃y ∈ ℝ : |f(y) − L| ≤ x ∧ ∀z ∈ ℝ{>y} : |f(z) − L| ≤ |f(x) − L|

Originally posted by humy
perhaps a better question is does either

2*∞ = ∞ is true

or

2*∞ > ∞ is false

make total sense in formal logic?

Unless I am mistaken, I have now worked out the answer to that question is 'no'.

I think I have worked out that to talk about equality or inequality of infinities of the same order is always nonsense.

Expressions such as 2*∞ = ∞ or 2*∞ > ∞ or even just ∞ = ∞ are neither true nor false but rather total nonsense!
To show why, lets consider defining two infinite sets of numbers (natural numbers in this case) where they are defined in exactly the same way except in the arbitrary label, label A or label B, they are given;

set A of numbers is 1, 2, 3, ...

set B of numbers is 1, 2, 3, ...

Now, for every number n in set A, there exists a corresponding number n in set B. This can be shown as;
1 → 1
2 → 2
3 → 3
...
So, because every element in A can be mapped to every element in B without any being left over, that would appear to give credence to there being an equal number of numbers in both sets and that to say ∞ = ∞ makes sense.

BUT, we also have; for every number n in set A, there exists a corresponding number 2*n in set B. This can be shown as;
1 → 2
2 → 4
3 → 6
...
So, because every element in A can be mapped to an element in B but there to be an infinite set of numbers of 1, 3, 5 ... left over in B that were not mapped to from A, that would appear to give credence to there being a greater number of numbers in B than in A thus implying ∞ < ∞ makes sense!

Similarly, we can apparently show ∞ > ∞ makes sense!

So now we have the paradox of ∞ = ∞ AND ∞ < ∞ AND ∞ > ∞ all apparently 'making sense' and that is not to mention that even ∞ < ∞ or ∞ > ∞ alone seems surely nonsense! I thus conclude none of these expressions makes sense, including the ∞ = ∞ one, thus solving this paradox.


Is this a valid argument?
And does it settle this issue once and for all i.e. proves ∞ = ∞ etc makes no sense?

Originally posted by humy
Is this a valid argument?
And does it settle this issue once and for all i.e. proves ∞ = ∞ etc makes no sense?

As I have said twice already, it depends on what you mean by ∞. There are certainly definitions under which the expressions may make sense. But, if you are taking ∞ to be a real number, as you apparently are, then yes, they don't make sense because there is no real number that accurately represents a count of all the integers.

This is similar to the fact that

1+2+3+4+... = -1/12

makes sense under particular definitions of the '=' sign but not others.

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

While looking at that page, I came across this page:
https://en.wikipedia.org/wiki/Monstrous_moonshine
which honestly goes way over my head.

Originally posted by twhitehead
[b But, if you are taking ∞ to be a real number, as you apparently are, ...[/b]

No, I am not.

As I have said twice already, it depends on what you mean by ∞. There are certainly definitions under which the expressions may make sense.

Can you give me an example of a definition of ∞ which would render my previous argument in that post invalid?

This is similar to the fact that

1+2+3+4+... = -1/12

No, it isn't a fact. We already covered this in another thread and I concluded this is wrong (and wrong under ANY definition of "=" ) period!

This was exactly what I said in my last post on that thread:

“....
I have been mulling over some of the links I have been given and finally noticed that one clearly implies that the infinite series:

1 + 2 + 3 + 4 …

cannot possibly equal -1/12 !

http://en.wikipedia.org/wiki/Divergent_series
“...In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. ...”

Thus, if the above statement is accurate and always true, the infinite series:

1 + 2 + 3 + 4 …


cannot possibly converge to -1/12 or anything finite since it is clear that the individual terms do not approach zero.

If the above statement in that link is accurate and always true, the same goes for the infinite series

1 – 1 + 1 – 1 + 1...

but, in this case, if I am interpreting the link correctly and that link is correct, it doesn't equal infinity but rather simply doesn't have a sum!

Thus, if I am interpreting the link correctly and that link is correct, the whole premise on that video link
YouTube
that claims to prove that the infinite sum 1 + 2 + 3 + 4 … equals -1/12 must be wrong (and is all a bit of nonsense ) because that video link says the infinite series 1 – 1 + 1 – 1 + 1... converges on ½ while the http://en.wikipedia.org/wiki/Divergent_series link implies it cannot converge because the individual terms don't tend to zero.

...”

Originally posted by humy
No, I am not.

I think you are:

So, because every element in A can be mapped to every element in B without any being left over, that would appear to give credence to there being an equal number of numbers in both sets and that to say ∞ = ∞ makes sense.

What does 'number' mean in the above?

No, it isn't a fact. We already covered this in another thread and I concluded this is wrong under ANY definition of "=", period!
You may have made that conclusion, but I certainly didn't. Also to claim something is wrong whatever the definition of its constituent parts is ridiculous.

Originally posted by twhitehead
I think you are:

So, because every element in A can be mapped to every element in B without any being left over, that would appear to give credence to there being an equal number of numbers in both sets and that to say ∞ = ∞ makes sense.

What does 'number' mean in the above?

[b]No, it isn't a fact. We already covered this in another t ...[text shortened]... Also to claim something is wrong whatever the definition of its constituent parts is ridiculous.

What does 'number' mean in the above?

in this narrow context only, either a real number or infinity. The operative word is 'or'. I am aware that, in conventional terminology 'infinity' is not a 'number'.

to claim something is wrong whatever the definition of its constituent parts is ridiculous.

Not ridiculous if one were to, as I meant to, exclude all ridiculous definitions.I would say 'ridiculous' definition would mean here one that the vast majority of people would strongly disagree with, such as "=" means "usually equivalent".

Originally posted by humy
Either a real number or infinity. The operative word is 'or'.

So this would make sense?:

So, because every element in A can be mapped to every element in B without any being left over, that would appear to give credence to there being an equal infinity of numbers in both sets and that to say ∞ = ∞ makes sense.

Not ridiculous if one were to, as I meant to, exclude all ridiculous definitions.I would say 'ridiculous' definition would mean here one that the vast majority of people would strongly disagree with.
But there are definitions of '=' for which it makes sense to the vast majority of mathematicians. I realise that the vast majority of people 'here' are not mathematicians, but given that the topic is mathematics, I don't think a democracy of non-mathematicians is a good idea.

It is actually less a case of '=' being redefined, but rather what we mean by the sum of an infinite series.

Originally posted by humy

What does 'number' mean in the above?

in this narrow context only, either a real number or infinity. The operative word is 'or'.

actually, I take that back. I mean " an infinity number " by 'number' there. I am aware that conventional maths terminology says 'infinity' is not a 'number' but don't know how to say it without either implying infinity is a real number or implying there exists infinity, which is a concept I am unsure of.
And yes, you were right in implying that that bit of what I said there didn't make sense.