[guest4091]

jeffreyH #41

The stick is not infinite in length since it terminates at one end.

The set of integers has a beginning and no end, and is considered 'infinite'.
I think you are confusing 'infinite' with 'eternal', without beginning and end, with respect to time.

This is always the problems with human conceptions of infinity.

I agree. The mind only experiences finite things, whether natural or manmade. Some only appear to be permanent due to a long decay/transformation process.
The stick;

Keeping it simple, align the zero of the ruler to the end of the stick. How do you perform the 2nd alignment of the stick with the ruler, if you can't find the other end? If you say it's a distance x units from the 1st end, you are wrong (since there is no largest integer x). I.e., if it has no 2nd end, it has no measurement. This exposes the faulty extrapolation of applying mathematical operations for finite objects to non-finite objects.
Cantor associated the idea of 'infinity' with God, and believed his thoughts concerning it were revelations from above. Given his chronic periods of depression, his concern with his standing in the mathematical community, and his assumed position as self appointed spokesman, his conclusions are not surprising.

This is just a continuation in human history of someone wanting to play god, but doesn't know how.

[Bill S (OP)]

A divine edict is not science because the supernatural is not recognized by the scientific method. Anything that seems Supernatural, has natural causes that we don’t yet understand.

I’m not suggesting that we should try to introduce the supernatural to scientific thought. I was simply wondering about the extent to which a possible violation of a hypothetical set of invariant natural laws might be scientific, rather than philosophical.

[jeffreyH]

I am not exactly confusing infinity with eternity since one could be considered spatial and the other temporal. We can't find an end to time since even vacuum fluctuations indicate changing states. So saying it's the big bang might be wrong.

[Bill S (OP)]

Where does the idea that the contiguous space has always existed “in a specific state”, come from? It has always existed and has always encompassed all there is, all matter, energy, everything, in one boundless, eternal, contiguous space, and the straight forward interpretation is that such a universe would be in more than one state, and would probably encompass all of the possible states permitted by the invariant natural laws.

Possibly a bad choice of wording on my part.  I was trying to look at infinity before and after a single change.

The states before and after that specific incident may have been states of continual change; but each would differ from the other as a result of the change we were considering.  Does that make sense?

[Bill S (OP)]

I would say that "infinity is not a specific number; it is larger than any specific number."

I’m reluctant to suggest you might be prevaricating, here, Evan; but the question was:

“Is infinity a number”.  Does answering it by saying that is is not a specific number mean that you think it is a number of some sort?  If so, what sort?

When you say: “it is larger than any specific number"; in what way is it larger?  Are you saying it is not a number, but has some numerical value?

[Bogie_smiles]

A divine edict is not science because the supernatural is not recognized by the scientific method. Anything that seems Supernatural, has natural causes that we don’t yet understand.

I’m not suggesting that we should try to introduce the supernatural to scientific thought. I was simply wondering about the extent to which a possible violation of a hypothetical set of invariant natural laws might be scientific, rather than philosophical.

Ok, I was just stating the obvious.

If a set of speculative and/or hypothetical invariant natural laws could be defined to make up a model, the premise would be that there can be no violation to that set. If any specific law identified as part of that model is violated, it falsifies the model.


My response is that the definition of a working concept of an infinite/eternal universe can be dissected by looking at individual natural laws, as I interpret you are suggesting, and showing that they can be violated.

For example, if the Universe could be shown to be finite spatially or temporally, it would violate the stated nature of the infinite/eternal universe in the definition, and would falsify the definition. The philosophy of the situation comes into play when the definition is not falsifiable, because there is no experiment to test it.

[Bill S (OP)]

I am not exactly confusing infinity with eternity since one could be considered spatial and the other temporal. We can't find an end to time since even vacuum fluctuations indicate changing states. So saying it's the big bang might be wrong.

“We can't find an end to time ”
 
This could be another of the difficulties that arise from the use of finite terminology in an attempt to discuss the infinite; but, just to clarify; are you saying that eternity is infinite time? 

[evan_au]

you might be prevaricating, here, Evan

Well, potentially, sometimes...

When you say: “it is larger than any specific number"; in what way is it larger?

For any specific, finite integer n, there exists a number n+1 which is larger than n.
Therefore n is less than infinity.

Mathematicians would write this as something like: ∀ n ∃ n+1: n+1 > n ∴ n < ∞
Where "∀" means "for all", and is another one of those mathematical tricks that allow you to make an infinite number of statements in a very small space.

  Are you saying it is not a number, but has some numerical value?

It cannot have a specific numerical value (as proven above).
(Perhaps prevaricating again): It is not a specific number, but is more of a trend. This trend is context-specific.

Mathematicians will say things like e^-x = 0
This means that the Limit (or trend) of e^-x is towards zero as x gets very large.

One of the reasons mathematicians avoid using ∞ as a number is that you can get yourself in trouble when you try to apply normal arithmetic/algebraic rules. For example, you could write: ∞ + ∞ = ∞

But when you reverse it and try to do ∞ - ∞, is the answer 0, or ∞? Or somewhere in-between?

Similarly, when you try and write ∞/∞, is the answer 0, 1, or ∞?
At times like these, you need to ask for more information.
In the example I gave earlier of Fibonnacci numbers,
F_n = ∞

You know this because for any finite integer y that you give me, there exists a finite value n whose Fibonnacci number F_n is greater than y.
∀ y ∃ n: F_n > y

Now, when you take the ratio of two successive Fibonnacci numbers:
F_n+1/F_n

It might look like ∞/∞, but that is only because you have thrown away most of the information.
In fact, when you consider all the information,
F_n+1/F_n = Φ ≈ 1.618033988749895...

“Is infinity a number”.  Does answering it by saying that is is not a specific number mean that you think it is a number of some sort?  If so, what sort?

I will probably get in trouble with real mathematicians here*, but provided you don't throw away all the information, you can sometimes come to some sort of answer. For example, if:
∞₁ = the number of positive integers = (1, 2, 3, 4, 5, 6, ....)
where "..." is another of those mathematical tricks for doing an infinite number of things on a single line
∞₂ = the number of positive even integers = (2, 4, 6, 8, ....)

You can show that ∞₁  = ∞₂, as described above
But what about ∞₁ - ∞₂? Is it 0, ∞, or somewhere in-between?

If you remove the set of positive even integers from the set of positive integers, you end with the set of the positive odd integers (which is an infinite set).

I venture to suggest that in this case, provided you don't throw away all the available information, it looks like you are doing ∞ - ∞ = ∞, but in reality you are subtracting infinite sets and counting the results, rather than subtracting infinite numbers and getting an answer.

However, different cases will produce different answers, because each individual case is different, when you consider all the information.

*I am only an imaginary mathematician (I only studied 3 years of maths at university level).

If a set of speculative and/or hypothetical invariant natural laws could be defined to make up a model, the premise would be that there can be no violation to that set. If any specific law identified as part of that model is violated, it falsifies the model.

We have talked in this thread about Cantor's mind-bending mathematical results.
 Kurt Gödel also had some mind-bending results. My simple interpretation is that in any sufficiently complex (mathematical) system, there will be true statements that you can't prove within that system, and/or there will be contradictions that you have to live with.

Mathematics does not map directly into the real world, but I think that your hypothetical natural invariant laws about the physical world may suffer the same fate.
See: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

[Bill S (OP)]

Thanks for that very expansive answer, Evan.  Please don’t think that I am unappreciative when I say that all you have done is confirm what I have said all along: namely, that mathematical infinities can be manipulated within the scope of mathematics. I am in no way qualified to criticise them or their use, even should I wish to do so. In fact, I have no wish to try.
 
What I am saying is that, as I see it, mathematical infinities and physical infinity are different things. There has been some fascinating stuff in this thread, but, as yet, nothing to convince me otherwise.   

[guest4091]

Cantor may have contributed to number theory, but his ideas about infinity were not accepted by all. The constructivist view rejects them.

The statement " there are as many even integers as integers",
is typically demonstrated using a 'one to one' correspondence as shown.
N: 1  2  3  4   5   6 ...
E: 2  4  6  8  10 12 ...
This is contradicted by:
1. Random sampling of integers results in an average of 50% even E, 50% odd D.
Statistics can be verified in the real world, and is useful in applications of probability.
2. In the above example, removing E from N leaves D, removing E form E leaves nothing, so where is the logic? An odd feature of this example is the appearance of the same integers in both sets.
The 'bijection' for example 1 defines y=2x, as a mapping from N to E. I see the results as not about the size of sets, but about the definition used for mapping.

Representing the first 'one to one/ correspondence above in a rectangular form, partitioned into subsets:

1   2   4   8...
3   6 12 24...
5 10 20 40...
...
The odd integers, all listed in column 1, are paired with the column of even integers to the right.
The remaining even integers in each column are paired with the column to the right.
The pairing is independent of direction. If a is paired with b, then b is paired with a.
The odd are paired 1 to 1 with an even.
The first subset of even (col 2) is paired with odd (col 1) and even (col 3).
The remaining even are paired with two columns.
Not all pairings are 1 to 1, and each integer appears only once.

A different correspondence can be formed with a subset of N as:
N: 2  4  6  8  10 12 ...
E: 2  4  6  8  10 12 ...
From this example, N is greater than E, since D in N has no corresponding members.

We can also map S the set of squares to N the set of naturals, without exhausting the set N.
N: 1  2  3  4  5  6  7  8  9 ...
S: 1           4                  9 ...
N contains S, but S does not contain N.
N>S

A true example of a 1-1 correspondence, "there are as many even integers as odd integers "
D: 1 3 5 7  9 11 ...
E: 2 4 6 8 10 12 ...

A subset of N can always be paired/mapped to itself and remain less than N.

Maybe the first statement of the correspondence is a poor choice of words, similar to ‘approaching infinity’.

[alancalverd]

A stick can indeed be infinite. Having one end is not a problem: however long your stick may be, I can conceive of a longer one, without limit. We often use the term "semi-infinite" in physics and applied maths to denote a quantity or space with an origin and no positive limit. It's just one of several useful infinities, and AFAIK there is provably an infinity of infinities for which we have no conceivable use.

There is no contradiction in a countable or "denumerable" infinity. All it means is that you can in principle assign an ordinal number to every member of the set, but that doesn't mean you can predict the  nth member

[hamdani yusuf]

It's an interesting video by Vsauce showing different types of infinity.

[guest4091]

infinite: adj
not measurable: without any limits that can be measured or realized
latin; not finite
It is an attribute modifying a noun, denoting a relationship to its surroundings, having no boundary.
infinity: noun
the property of (an unidentified) something being infinite.

Assume a substance s is infinite. Here is a volume of space with no s. If this space is filled with s, then the total volume of s is increased. This is a contradiction since by definition s is unlimited.
There cannot be an infinite amount of any substance.

[guest4091]

A stick can indeed be infinite. Having one end is not a problem: however long your stick may be, I can conceive of a longer one, without limit.

How can you extend something without a limit?
(infinite: unmeasureable).
A dictionary is the best reference.

[guest4091]

The question is: how many square integers are there in the interval from 1 to 10?
If we count the squares, there are 10.
The one to one correspondence states there are as many squares as integers.
N: 1  2  3   4...  9   10
S:  1  4  9 16...81 100
What is it that results in that statement?
Yes, there are 10 integers in each row, but the integers are symbols representing sets.
There are 10 symbols in each row, so counting from '1' in the ordered set N, we arrive at the symbol that represents the set with the correct number of elements in the set S.
The 1-1 counts symbols, not sets.
The set of 10 squares S, ranges from 1 to 100, a density of 1/10, meaning there are 90 integers in N from 1 to 100 that are not squares.
Cantors interpretation is not acceptable.

[alancalverd]

A stick can indeed be infinite. Having one end is not a problem: however long your stick may be, I can conceive of a longer one, without limit.

How can you extend something without a limit?
(infinite: unmeasureable).
A dictionary is the best reference.

You don't have to extend the stick, but just consider the possibility of adding a bit at either end, to whatever stick the other guy is holding. Now you have a concept of a seminfinite body.

A dictionary is a poor reference for such a technical term as infinity.

Consider  first the number of possible integers 1... N ... N+1 etc.However large you choose N, I can propose N+1 as a larger integer, so the number of possible integers is infinite, even by a dictionary definition.

But between any two integers there is an infinite number of rational numbers such as 2, 3/2, 4/3,.....(N + 1)/N....., 1 So the number of rationals is a greater infinity than the number of integers.

And between any pair of rational numbers there is an infinity of irrationals, so there is yet a greater infinity........

One question worth asking is whether the number of irrationals is a denumerable infinity. Now we are talking maths, not dictionaries. Fun, isn't it?

[evan_au]

1. Random sampling of integers results in an average of 50% even E, 50% odd D.
Statistics can be verified in the real world, and is useful in applications of probability.

How often have you run into an infinite number of real objects in the real world?
Conventional Probability is of limited use when dealing with infinities.
50% of ∞ = ∞
1% of ∞ = ∞

Unless you get down to infinitesimal probabilities, which are numbers which are as small as ∞ is large; then you can come up with finite answers that start to follow conventional relationships like < and >. But you have to be careful not to throw away too much information, or you end up with statements like 0 x ∞, which you can't resolve.

Integration adds up an infinite quantity of infinitesimals to come up with a finite answer - at least, in the examples they give you in high school. Not all functions are so forgiving.
https://en.wikipedia.org/wiki/Infinitesimal

But between any two integers there is an infinite number of rational numbers such as 1/2, 3/2, 4/3,.....(N + 1)/N....., 1 So the number of rationals is a greater infinity than the number of integers.

Cantor showed a way of mapping the fractions onto the integers - this is shown nicely in the first couple of minutes of the video. So there is the same number of rationals as there are integers = .ℵ₀

Mind-bending stuff!
Cantor showed that  the number of real numbers (which includes irrationals) = ℵ₁ is greater than the number of integers = ℵ₀. So the real numbers is not countable, since there are more of them than there are counting numbers.

[alancalverd]

In fact, random sampling of a set of integers won't give you 50% evens. If it did, you would suspect your sampling process of being selective, not random. It would be fairly clear as the sampling progressed that it was tending towards 50%, but if the sample is of an odd number of integers it cannot contain 50% of evens.

As an aside: We can use the χ² "goodness of fit" parameter to determine whether we have a truly random sample of an experimental result. If we obtain an exact fit to the hypothesis at each measurement, it is more likely than not that the equipment is faulty and biased. Simplest case is where the expectation value is zero (a null experiment) and successive readings are exactly zero. The probability is that the equipment is not working at all. Not a joke: you can fly "perfectly straight and level" and spiral into the sea because the gyroscopic instruments are not switched on!

[Bill S (OP)]

I'm just home from a short stay in Hosp.  I'll be back in the thread as soon as I feel a bit better, and can catch up.

Don't let it die in the meantime.   Looks as though there is some interesting content.

[guest4091]

infinite: adj
not measurable: without any limits that can be measured or realized
latin; not finite
It is an attribute modifying a noun, denoting a relationship to its surroundings, having no boundary.
infinity: noun
(an unidentified) something being infinite.

Assume a substance s is infinite. Here is a volume of space with no s. If this space is filled with s, then the total volume of s is increased. This is a contradiction since by definition s is unlimited.
There cannot be an infinite amount of any substance.
The universe shows us just the opposite. Matter is discrete finite units at all levels, from astronomical to atomic. The closest candidate to qualify as infinite would be space.
Even that isn't necessary since physics at the local level is independent of remote mass (relativity).

That’s my basis for no usable/meaningful application of a physical ‘infinity’.