- Infinity does not exist - see the OP - so this whole conversation is about MARSH GAS

- Even if infinity existed, finity is a subset of infinity and what works for infinity needs to work for finity also. One-to-one correspondence does not work for both so it is flawed

- You are very closed minded. You think that maths has it 100% correct - that is an unbelievably naive assumption to make. At what point in the past was human knowledge 100% correct? At what point in the future will it be 100% correct? It was not, is not and will never be 100% correct. The blind (=you) swallow everything without question. I question things. Now I may be wrong but at least I keep an open mind rather than just dumbly reciting the received 'wisdom'.

When you get down to the nitty gritty and use numbers, as in a computer program, they are all rational and thus countable. So how many rational numbers (fractions) do you think lie in the interval [0,1]? :chin: — John Gill

Depends on the hardware/software you are using I guess. Computers use discrete binary representations of numbers. Computers are real. Reality is real. Continua maybe just a figment of our imagination as far as reality goes. Time will tell.

2) Finite means definite. But it's only definite for a particular value. But no particular value can be assigned, therefore not finite. — tim wood

As you can imagine I am not a believer in the possibility of continua occurring in nature and I am not alone in this belief (eg loop quantum gravity). If spacetime is actually discrete, then case 2 above will have a particular, assignable, finite width. Time will tell. I'd point out that we will never, ever be able to empirically prove spacetime is continuous, but we may be able to prove it is discrete.

What if the counting will go on forever - not a trivial problem. Answer: you define counting and establish some rules for it, then you count and if the count is even at every point, then the two things are same size. — tim wood

Yes and the rules set theory defines are broken. If something goes on forever, you can't count it - even with an infinity of time it is not possible to measure something that goes on forever. This is what Galileo recognised and what Cantor ignored - and it leads to spurious results, such as the number of naturals is the same as the number of rationals - how can anyone swallow that? For each natural, there is an infinite number of rationals... One-to-one correspondence gives nonsense results.

There are only three possibilities. A number of the real number line must have:

1. A zero width
2. A non-zero width
3. A undefined width

So how many reals between 0 and 1?

1. 1/0=UNDEFINED
2. 1/(some finite small number)=(some finite large number)
3. 1/UNDEFINED = UNDEFINED

I believe that a number is a purely imaginary concept so we can imagine it to have zero width so definition 1 above is what we do in our minds. This is also consistent with the definition of a point in maths as having zero extent - so we can imagine a point corresponding to a number on the real number line.

Now you could claim that a number has an infinitesimal width. But that is 1/∞ and in my opinion ∞ and its inverse are not well defined - the point of the whole OP.

Numbers exist in our minds only so we can imagine a zero width number and an UNDEFINED number of numbers in a finite interval.

If you were to try to represent numbers in the reality external to your mind, you would find that any line used to represent the real number line would be composed of molecules so there would be a finite number of numbers on a real line segment.

God be praised - a counter argument from @Tim Wood - a miracle!

In each finite segment of the naturals, the number of naturals is approximately twice that of the even naturals. Yet each natural can be doubled to give a one-to-one correspondence to the even naturals. This is Galileo's paradox - by one measure, the cardinality of the two sets is different, by another measure its the same.

What Cantor did was to ignore half the paradox and define 'size' in terms purely in terms of one-to-one correspondence. That is expressing an opinion on the nature of size that is incompatible with all finite subsets of the naturals - so yes MATHS IS OPINION - and in my opinion Cantor got it wrong.

Infinity is by definition unmeasurable so it has no size - how can you assign a measure to something that only exists in our minds and our minds says goes on forever?

If you disagree, tell me the size of the naturals. And no alpeh-zero is not the answer - that's a symbol without meaning.

:grimace: You are full of s**t. If you had any counter arguments to my points you would post them... but you don't so shut up.

OK I get you:

https://www.compoundchem.com/2016/03/22/boiling-point/

That's right! Maths is nothing if not opinion. Btw, do you have an opinion as to the temperature that water boils? — tim wood

100 °C

What is your point exactly?

There is an infinity between the numbers one and two because numbers are infinitely divisible. — Amergin

This point in a slightly different guise, is discussed at length on another thread:

https://thephilosophyforum.com/discussion/comment/362707

In short (and my opinion only) - a number has zero width, so the number of numbers between 1 and 2 is 1/0=UNDEFINED and not infinity.

If a number has non-zero width, that leads to a finite number of numbers between 1 and 2.

This is known as Galileo paradox:

https://en.wikipedia.org/wiki/Galileo%27s_paradox

I share Galileo's rather than Cantor's opinion, but I am in the minority.

I think on step 6 above, saying 1*∞=∞ is a circular definition so is not too enlightening.

I think my doubts are around how infinity is defined. Limits seem to do a good job of defining potential infinity but how is actual infinity defined? Set theory defines 'countable actual infinity' as the the 'cardinality of the set of natural numbers' - which is about as meaningless a definition as could be imagined.

Actual infinity is not computable and not (it seems) definable, leading to a suspicion that it cannot be a real concept - it may exist only in our minds (along with talking trees and square circles) and is therefore not realisable.

Maybe @JohnDoe, @softwhere or @John Gill have opinions?

Do you suppose there's a reason why points are zero-dimensional? — TheMadFool

I find the concept of a dimensionless object difficult - it has no extents so it cannot have any existence - how can any sound reasoning performed with a non-existent object - assuming its existence (in order to reason with it) leads straight to a contradiction?

How would we define distance? The beginning/end of one point to the beginning/end of another point? Why not just consider the beginning/end as zero-dimensional points? — TheMadFool

Well the concept of 'measure', as alluded to above by @softwhere, seems to be math's answer. But measure theory does not seem (from my very limited knowledge of it - ?) to provide a justification for treating a point as dimensionless (or that there are infinite points on a line segment).

To define distance in a conventional sense is to have a unit of measure and a zero dimensional point is not a valid unit of measure - if we say a point has zero length and try to use it as a measure, then the measure of all line segments, no mater what length, is UNDEFINED.

I'm not even sure it is correct to say the beginning/end of a line are points - points don't exist - so would the line even have a beginning/end?

That seems like a true assessment. Why do you find it unattractive that there might be somebody far away that is just like you? — DanielP

I find the notion that there are an actually infinite number of identical me/planet earth/this galaxy in time and/or space to be absurd, so I tend to regard the argument I gave as Reductio ad absurdum.

Spacetime began with the BB 14 billion years ago and has been expanding ever since at a finite rate - so spacetime is logically finite.

If you could sit in an armchair and watch the BB unfold, what would you observe? Well GR says that time is observed to run slower in a gravitational field and we have experimental evidence that supports the theoretical claim, so:

1. One billion years after the BB, matter density is not so high, so time is observed to run relatively quickly.
2. One million years after the BB, matter density is higher so time is observed to run slower
3. One second after the BB, matter density is very high, so time seems to be almost at a standstill
4. At the moment of the BB, no-one knows what happens to time, but you can see the pattern.

A whole lot of nothing I’m afraid. The being you could just as easily have said ‘art is the meaning of life, because I’m viewing everything as art’. — I like sushi

Art is a subclass/subtype of information?

But the vast bulk of the universe is devoid of information — Wayfarer

Then what is astronomy?

Fields pervade the universe such as the Higgs field etc... these fields fluctuate in line with the uncertainty principle... virtual particles are constantly created and destroyed. So even empty space is information rich.

Sorry I should of said:

- A point has zero length according to maths definition
- But according to my intuition, a point must have non-zero length

If you live in the city, consider the tall structures of concrete and steel. Why is it that they don't tumble down? Aren't they based on the axiom of infinity? No. Formal set theory is arguably more of an aesthetic enterprise. — softwhere

I would contend that set theory gives an unwarranted legitimacy to actual infinity which influences the physical sciences. Some cosmologists are obsessed with actually infinite time and space and they lean on concepts borrowed from pure maths to justify such infinite models.

I will humor you. The number of points in [0,1] is uncountably infinite. — softwhere

Measure theory does not seems to provide any justification for the above claim - neither do I see any justification anywhere else in maths.

I understand where you are going. But how sensible is this time before time? I find it as questionable as intuitions of actual infinity. Personally I think human cognition runs aground on issues like this. — softwhere

There are a number of good arguments that all point to a start of time (one example: perpetual motion is impossible, things are currently in motion -> a start of time). If there is another type of 'time' 'before' our time, the same arguments apply to that 2nd type of time - it must also have a start. Obviously there cannot be an infinite regress of such times, therefore we seem to be left only with the possibility that something must exist that is timeless /atemporal.

You maybe right stating 'human cognition runs aground on issues like this'. Philosophers have grappled with the nature of timelessness for centuries and I have not yet encountered a satisfactory explanation of what it could be.

I imagine our universe as a 2d space time diagram - a plane of finite dimensions. Then I imagine a point off to the side of this plane that represents a timeless thing. Then I imagine there is a mapping from the point off to the side to each point in the plane - the timeless thing experiences everything in time in one 'eternal now' and can interact with anything in spacetime. Of course this does not really shed any light on the nature of the point - the timeless thing.

There, when dealing with limits, it is convenient to pretend that there exist two points ∞ and -∞ which are endpoints of the real line. Then ∞+∞=∞, and all other formal rules makes it easier to deal with limits without worrying much about particular cases of infinite limit. — John Gill

We are talking about the nature of time, whether it has a beginning or end specifically. Such a conversation is intimately linked to the existence or non-existence of Actual Infinity. Maths treatment of the subject could hardly be described as definitive - a set of non-sensical assumptions IMO. Notice I have highlighted the phases 'pretend', 'without worrying much'... such words hardly inspire confidence...

There is no largest number - numbers go on for ever in each direction - there is no such number as ∞ - so a mapping of non-existent points at actual infinity to the north pole of a sphere is a nonsense procedure.

And yes z/0=∞ appears to have 'valid' (!) applications in maths:

"The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=∞ well-behaved."

- https://en.wikipedia.org/wiki/Riemann_sphere

"There are, however, contexts in which division by zero can be considered as defined. For example, division by zero z/0 for z in C^*!=0 in the extended complex plane C-* is defined to be a quantity known as complex infinity.”

http://mathworld.wolfram.com/DivisionbyZero.html

“Nonetheless, it is customary to define division on C ∪ {∞} by
z/0 = ∞ and z/∞ =0”

https://en.wikipedia.org/wiki/Riemann_sphere#Extended_complex_numbers

As you are both mathematicians, I hope you don’t mind if I take the opportunity to explain my concern a bit further. I feel there are some deep problems with infinity in the foundations of mathematics and it seems to me that many mathematicians are very invested in their discipline and their hard-won knowledge - so that they are not usually eager to confront such issues.

For example, take the extended complex numbers - the set of complex numbers plus ∞. The definition used for ∞ is z/0=∞. Now you can call that ‘an assumption’ if you like (and a pseudo-justification in terms of limits can be given) but it is plainly a wrong assumption. I believe there are then fields of maths (like complex analysis) which build on the idea of the extended complex numbers. Then people in the physical sciences build further theories based on these ideas. The net result is whole vertical slices of human ‘knowledge’ which are based on wrong assumptions and are therefore not valid knowledge.

Similar bad assumptions to the above example can be found in the hyperreal numbers and the projectively extended real line. Another example, already discussed above, is the axiom of infinity from set theory - the assumption of the existence of actually infinite sets of objects. It is a bad assumption to make and set theory is based on that bad assumption. Many things in maths and science are then built upon the foundation of set theory. Again we have whole swaths of knowledge based on bad assumptions - all that ‘knowledge’ is therefore not valid.

I am not a mathematician so I cannot call out more examples than this, but I’d be surprised if there are not more. It is therefore not surprising (I hope) that laymen such as myself are disinclined to try learning more of advanced mathematics - my (admittedly limited) experience of such is that (what can happen) is early on, in the foundations, a bad assumption is made and then a body of interesting, complex but ultimately invalid results are derived based on that bad assumption.

I feel mathematics has a responsibility to the rest of the physical sciences to keep the assumptions reasonable. By reasonable, I include assumptions that are provisional - they may lead to interesting, but provisional results (eg non-euclidean geometry). I also class as reasonable assumptions that widen the scope of traditional mathematics, such as the introduction of new types of numbers (eg complex numbers). But assumptions that are plain wrong/bad (counter logical) lead nowhere useful, lead other folks (in the physical sciences) astray, and result in lots of clever folk wasting huge amounts of time on wild goose chases (eg a good portion of modern cosmology is like this IMO).

Your thoughts?

Agreed. A 0-dimensional point can in no way be the constituent of a 1-dimensional line segment - the point has zero length and the line segment has non-zero length. So it is incorrect to say, as mathematicians often do, that a line segment contains an actual infinity of points.

Likewise, a 1-dimensional line cannot be the constituent of a 2-dimensional plain - the line has length but zero width so it cannot be the 'parts' of a plain (which has non-zero length and non-zero width).

An infinite universe results in strange phenomena such as:

https://thephilosophyforum.com/discussion/comment/360596

That is to say, everything that is must logically repeat itself in time and in space - go forward or back in time far enough and there will be an identical copy of you, the earth, this whole galaxy. Likewise go left or right far enough in space and you will find similar identical copies. In fact an infinite number of copies in both cases. This is one of the reasons why I personally do not believe in actual infinity or an infinite universe.

And (the) grounds for thinking the universe was "created"? — 180 Proof

See this OP:

https://thephilosophyforum.com/discussion/6218/the-universe-cannot-have-existed-forever/p1

Amazingly, if you haven't already noticed, there is a rather inexplicable absence of information in things that matter the most:

1. Meaning of life
2. How to live the good life
3. God
4. Afterlife (death)
5. Morality
6. The theory of everything
7. Consciousness
8. Origin of life — TheMadFool

1 and 2 - I feel these are addressed in the OP?

3 and 4 - Seem to relate more to the meaning of death than life?

5 - There appears to be a link between morality and information, see: https://thephilosophyforum.com/discussion/comment/363177

6 - By which you mean a theory that extends QM to include gravity? QM seems to me to be about information (particles) and our ability to measure information (uncertainty principle).

7 - Consciousness seems to be the processing of information as opposed to unconsciousness which is the complete lack of processing of information - the senses (information sources) are deactivated and the mind (information processor) is inactive.

8 - I feel that DNA is information, so the concept of information and the origins of life are intertwined.

I feel you are avoiding addressing my point and attempting to blind me with references to advanced math. If advanced math disagrees with basic math / basic logic then advanced math must contain logic errors.

The measure of the interval [0,1] is 1 and the measure of the interval [0,2] is 2. This way of classifying size also leads to the conclusion that a point must have non-zero length:

length of a interval = pointsize * pointnumber

Neither of 'pointsize' and 'pointnumber' can be zero because then the measure of the two intervals would be equal (zero in both cases). So a point cannot have zero length.

?

I still feel maths does not currently have a complete understanding of infinite series:

- At any intermediate point in the evaluation of Grandi's series, it always has a sum of 1 or 0. Therefore logically the final sum can only be 1 or 0 - there are no other possibilities
- But mathematical methods for evaluating series yield the sum of 1/2
- Maths calls the series divergent as the individual terms do not approach zero
- But if we knew whether ∞ is odd/even we could evaluate the series
- But my contention is there is no such thing as actual ∞
- So IMO the final sum of a series (taken to ∞) is a meaningless concept (in some instances)

This has bothered me since you first brought it up not a while ago. I'm not a mathematician but 1 here is a length and when you divide a length you don't get a point. What you get is another length.

Also, a point isn't defined in terms of how big/small it is i.e. it isn't dependent for its existence on its own dimensions which as you rightly pointed out is zero. A point is actually defined in terms of its distance from the origin (0,0) or some other reference point. — TheMadFool

OK, so your interpretation is (as I understand it) that that a line segment is not composed of infinite points, but is composed of sub-lengths. I am in agreement. I would point out that the length of a sub-length cannot be zero else all line segments would have the same size.

Imagine three galaxies in infinite space A, B, and C. Suppose the distance between them is 4,000 lightyears. Can't the space between these galaxies increase, not because they're moving but because space is being created between them. In other words I see a possibility of an infinite and expanding space. — TheMadFool

If the distance between them is currently 4000 ly and the universe is expanding. then there must have been a time in the past when the distance between them was 3000 ly, 2000 ly, 1000 ly, 0 ly. At the final point, when the galaxies are co-located, the universe cannot be expanding. So I think that infinite expansion is impossible. I believe there are some cosmologists who disagree with me on this.

What about time itself? Did it have a beginning? If space can be infinite and time is "just another" dimension, and if space can be infinite can't time be too? — TheMadFool

It comes to a question of origins. I believe that there must have been a first cause for everything in time (the cause of the BB probably). Then the obvious question is what caused the 'first cause'. We could answer that by introducing another cause to cause the 'first cause', but then we would need another cause, and another, so we end up with an infinite regress with no ultimate first cause of everything - which is impossible.

So there seems to be a need for a first cause and there cannot be an empty stretch of infinite time preceding the first cause - then there would be nothing but emptiness to cause the first cause - which is impossible.

So there seems to be a need for an 'uncaused first cause'. How do you get an 'uncaused cause'? Well causality is a feature of time, so placing the first cause beyond time seems to be the only way to have an 'uncaused cause' - then there is nothing logically or sequentially 'before' the first cause - the first cause has permanent uncaused existence. The first cause is then the cause of / creator of time (time must have a start).

I can sum up the argument with an altered version of the PSR:

- Everything in time must have a cause

Which leads naturally to a timeless first cause.

This also leads to, incidentally, an answer to 'why is there something rather than nothing?' - there has always been something - uncaused and beyond time - there is simply no 'why?' for something that is uncaused.

That seems consistent. I'll keep testing your hypothesis. Let's say a person did not kill people like Hitler, but tortured people a lot and made them miserable. He also lied, cheated, and kicked puppies in the face. I would imagine it reasonable to call this person a bad person, and yet no information was lost in this case. — Samuel Lacrampe

Interesting point. I think it could maybe be argued that the unhappiness that results from such behaviour causes a reduction in the amount of information produced - people who are down in the dumps/unhappy/depressed generate less (high quality) information than happy people?

By that definition, an elderly computer will have lived a happy meaningful life. — Gnomon

Fair point. So I qualify my original statement with: Something has to have life in order for the meaning of life to be applicable.

The Enformationism Worldview : http://enformationism.info/enformationism.info/ — Gnomon

That's an interesting web site you have! I am having a browse through it.

Basically they had to have it if they wanted the natural numbers, and they had to have the natural numbers. But others have wanted to take the natural numbers as fundamental. — softwhere

The problem I see is that (applied) mathematics forms the basis for our understanding of reality. So scientists pick up definitions and theories from maths and apply them to the physical sciences.

Now the set of natural number exists purely in our minds - my believe is there is nothing in reality akin to it. So there is this impossible concept which is taken from maths and is being applied in the physical sciences - producing erroneous results - cosmology is the biggest offender.

I feel a way of defining the natural numbers without resorting to questionable concepts like actual infinity would be the way to go.

Also, choice #1 does not imply that 1/0 = infinity. Saying so is pseudo-math. — softwhere

It does imply that 1/0 = ∞, we need only pre-school maths to arrive at such a conclusion:

1. Maths claims an infinite/uncountable number of points on a line segment
2. A point is defined by maths to have length 0
3. The line segment in question is length 1
4. So we must divide the interval into 0 equal pieces to find the number of points in it
5. Hence the number of points is 1/0 = ∞

Where is your dispute with the above reasoning?

So what if unchecked information processing actually causes our extinction? — Nils Loc

So you mean some sort of Terminator/Matrix scenario where the machines take over and enslave us all? If we ever develop AI, I think it will value information too and thus treat humans as a source of information and respect them accordingly.

Is there anything inherent difference in human (anthropocentric) processing of information (as if we had free will to do anything else) and the kind that occurs in the natural world for slime mold, trees and colliding galaxies? — Nils Loc

I guess I was trying to get at the meaning of life from the perspective of an intelligent entity (be it an organic life form, an AI or a deity). I am not disputing that information processing continues in the universe, regardless if there is anything to observe and value it.

Slime molds do actually seem to be quite information-savvy:

http://www.abc.net.au/science/articles/2000/09/28/189608.htm?site=galileo&topic=latest

If the purpose of a person was merely to gather and produce information, then anybody that does that should be called a "good person". But that is absurd. I'm sure Hitler gathered and produced as much information as any other person (if not more, being that he is famous), but he is nearly-universally judged to be a bad person. — Samuel Lacrampe

Hitler was a destroyer of people - our primary source of information - and therefore a destroyer of information - therefore I think I he can be classified as a bad person.

So I think the definition of good / bad person in informational terms is actually recursive: There is the amount of information you produce yourself, but in addition, if you save many lives, then recursively that results in a large amount of additional information. And if, as with Hitler, if you end many lives, then recursively that results in the destruction of large amounts of information.

I think that the intelligence behind the creation of the universe must be from beyond spacetime - it created spacetime so it must be external to spacetime.

In that sense maybe it does not have existence in the manner that we understand conventionally, but it must exist in the sense that it must be able to interact in some way with matter/energy/spacetime else it could not of created such.

So I don't believe in a purely spiritual creator of the universe - it is not possible to wish/think/shout a universe into existence.

To me this frames infinity as an object that already exists, already has a nature. Philosophers can compare their intuitions in natural language, but mathematicians have got to make some rules. — softwhere

But as I understand it, maths frames infinity as an object that already exists (axiom of infinity). I believe that axioms should be more than assumptions - they should be self-evident truths - and what is self-evident about the existence of an actually infinite set? Parallel lines not meeting I can swallow, but an actually infinite set?

So perhaps the burden is on Cantor's critics to offer a mathematical substitute. — softwhere

I am not a mathematician but I would imagine that the vital areas of mathematics (IE calculus) could function well enough using just the concept of potential infinity. I do not see why the concept of actual infinity is needed - it just leads to paradoxes.

You misunderstand me. The measure of a set is different than its cadinality. — softwhere

I am not disputing it is possible to measure intervals, I am disputing the common mathematical claim that there is an actually infinity of points on a line segment length 1.

How many points do you claim there are on a line segment length 1? The answer must logically be one of the following:

1. Infinite number
2. Finite number
3. Undefined
(there are no other possibilities)

If it is [1], that means 1/0=∞ which is nonsense
If it is [2], then a point must have non-zero length which is not the definition used in maths.
So I contend it must be [3].

God is not, IMO, the God of ancient scriptures such as the Bible etc... The 3O's and various other capabilities traditionally attributed to God are logically disprovable.

My opinion is that there is probably is an intelligence behind the creation of universe, but it hard to classify its exact nature.

I know very little about anti-realism. From your anti-realist perspective, what exactly is the past?

To me the past is a deducible concept without referencing external realities - I have thoughts, these thoughts from a causal chain. The present exists, there are thoughts that I am no longer having, so the past exists. There are thoughts that I will be having so the future exists. I can label each thought with an integer. Assuming a past eternity, then the number of thoughts would be equal to the highest number. But there is no highest number, so a past eternity is impossible?

The cardinality of a set depends on the notion of bijection — softwhere

Bijection/one-to-one correspondence is a procedure that produces paradoxes like Galileo's Paradox, or the cardinality of the naturals is the same as the cardinality of the rationals. It is therefore to my mind an unsound procedure. Cantor did nothing to help our understanding of infinity IMO; he has lead us down the wrong path entirely.

Hence the measure of an uncountable union of points... — softwhere

My (and Galileo's) point exactly - you fundamentally cannot measure something that is
uncountable/infinite - you would never finish measuring it - it is impossible to measure and claiming that bijection can provide a sound measure is ignoring the evidence (of paradoxes).

1/0 undefined. Suggest you move on to another topic. "infinite sum" is OK amongst professional mathematicians. — John Gill

The infinite sum concept in maths has definite problems, see here for an example:

https://en.wikipedia.org/wiki/Thomson%27s_lamp

Thanks Christian! It is getting late and I'm getting nowhere with @Banno so I guess I will say good night to everyone!

You've set up a false dilemma. — Banno

How so?

You state that 1/0 is illegitimate.

You also agree with the mathematical definition of a point as having zero extent.

Then the question of how many points there are on a line segment length one is perfectly valid:

- If its uncountable/infinite, then that suggests that 1/0 is legitimate
- If its undefined, then you are agreeing with me

Which do you choose?

Thats typical... don't agree with someone... let's try to get them banned. You'd like a forum where everyone agrees with you. What exactly would be the point of such a forum?