You will see in the OP that I did state it was two sets of identical (IE unnumbered) bananas.

The fact that points have no size (which, by the way, does NOT mean that points have zero-size), whereas line segments (whether continuous or discrete) do, does not mean that line segments are not made out of points. — Magnus Anderson

If you check the OP, I did consider the possibility that points have no size. That leads to the size of a point being UNDEFINED and all line segments having an UNDEFINED length.

You don't measure the length of a line segment by counting how many points it has, you measure it by counting how many pairs of points-at-certain-distance it has. "This line is 10cm long" means "This line is made out of 10 pairs of points-at-1cm-from-each-other". — Magnus Anderson

You establish above that your line segment is made out of 10, 1cm sub-segments. IE if we switch to discreetly/finitely sized sub-segments (/points), suddenly we can come up with a meaningful definition of length. That should tell you something - with discrete/finite sized points/sub-segments, lo and behold, the maths suddenly starts making sense. It is when we use non-sensical definitions like ‘points have no size’ that we find the maths always leads to contradictions.

But it is made out of points. It's just that the length of a line is not measured the way Devans99 thinks it is measured. You don't measure the length of a line by summing the lengths of its smallest parts (which are points.) Points have no length. They do not have such a property. Length is something that exists between two points. In order to measure the length of a line you must count the number of pairs of points-at-a-certain-distance that constitute it. — Magnus Anderson

You cannot simultaneously hold that line segments (which have length) are made out of points and points have no length - that’s a plain contradiction.

Most people define existence (not necessarily verbally but certainly intuitively) in such a way that even things that have no length (such as points, colors, sounds, etc) can be said to exist. — Magnus Anderson

Colours have a wavelength, so do sounds so they can be said to have existence. Points however, defined as having no extent, clearly cannot not exist. A line segment has extent so can be said to have existence. Maths, claiming that non-existent things can be the constituents of existing things, is tying itself in a logical knot.

Slight correction, they do reference Euclid's non-sensical definition of a point: "a point is that which has no part" on page 27.

I found the book you are referring to here:

https://download.tuxfamily.org/openmathdep/euclid/Euclid_and_Beyond-Hartshorne.pdf

It is 500 pages long and the words continua and continuum are never mentioned. It talks about 'points' a lot without even defining the term as far as I can see.

?

I was happy as a child, without a baseline of suffering. I can conceive of a system where people are happy, and they consciously realize that, without having had to suffer, or even without having a concept of suffering.

Why couldn't God create that? He is not omnipotent? — god must be atheist

But people would be happier still if they have had that initial exposure to evil (so that they have a way of measuring/quantifying good). My point is that, yes, there are many 'good' answers to the question, but the only optimal answer is that happiness is maximised in those who have already experienced evil.

Why not go from extremely good to superbly good, to ecstatically good? Why start at a level below acceptable? — god must be atheist

The lower you start, the more effective the regime is at maximising happiness over the long term (in a purely mathematical sense). Of course that is an argument for starting everyone of in hell, but I think that this would not be acceptable, based on a human decency argument. It also might lead to long term trauma in individuals. So some evil but not an excessive amount of evil seems right.

That lowers the bar of "omnipotent" below the acceptable level of the definition. — god must be atheist

But if any mind is a logic processor plus memory then even an omnipotent God can only create minds of a similar nature. Your point is similar to 'why can't God create square circles?' - an omnipotent God can only perform things that are logically possible. A mind without memory is not a mind as we know it and its the memory of evil that make the good times good.

Thanks for the reply! As you no doubt realise I am sort of playing devil's advocate here...

... my objection to your answer to the problem of evil is that it hinges on the human mind working a certain way, and God would have the power to make it not work that way, and so not require suffering on Earth. — Pfhorrest

But a mind - an information processing machine with a memory - may only be able to take on a singular, fundamental form. For example: all our computers are similar in nature and mimic the human mind. It maybe that creating an intelligent entity that is fundamentally different from ourselves is impossible even for an omnipotent God? 'Great minds think alike' is the saying - I would modify that to: 'All minds think alike'.

The basic law of 'in order to appreciate a maximum, one has to experience a minimum' may hold for all possible intelligent life forms and it maybe beyond the powers of an omnipotent God to change this basic reality.

Also consider that here on Earth people tend back toward a base level of happiness regardless of their circumstances, and some people can go back to being happy even after a horrible tragedy that they still nominally suffer from, while others can go back to being miserable even after they hit the jackpot and solved all of their nominal problems — Pfhorrest

So people may need a 'refresher course' in evil in order to continue to fully appreciate heaven? I would have thought that the forward thinking dread that would instil in people would be sufficient to nullify any beneficial effects on total net happiness? I think instead it maybe best physiologically to get the worst bit over with first - the experience of evil on earth - followed by the good bit (heaven).

The solution to both of those is to make human minds work differently, make people more inclined to be happy with and interested in whatever they currently have indefinitely and not get bored or tired of it and sink into bad feelings for no good reason. — Pfhorrest

If that was possible, God would only do it for a mind in heaven: suffering is required to define a baseline for happiness and in order to value happiness.

Aristotle was wrong to say that an object is potentially divisible but not actually so, because something contains the same volume whether divided or not. — Gregory

There is a distinction between:

1. The process of division - we cannot go on dividing an object forever because we would never finish the division process - so we can not take an actual object and make an actual infinity of pieces out of it
2. Something that exists and was never created could exist in a form that is already divided infinitely

So it seems it is impossible to create/manufacture the actually infinite, but actually infinite things could exist - uncreated things - as some hold spacetime to be uncreated (not my view).

- Yet the Dedekind-Cantor continuum is taught in school along with the fact that a point has zero width. So my objections are bang upto date, as far as I can see.
- No-one has yet pointed out any logic/math error in my OP.
- If I have something wrong, then someone should set me straight, rather than vague hand waving
- At least a link to your preferred definition of the continuum would be nice

This is a very interesting video.

If anything is infinite, it is our lack of knowledge.

That does not mean that every "proposition regarded as self-evidently true without proof" is an axiom. — JeffJo

Perhaps you can point to an example of a "proposition regarded as self-evidently true without proof' that is not an axiom?

So once again, the statement you claimed was an axiom was never stated as part of such a set, from which theorems could be derived. It was not an axiom, it was a near-religious belief. — JeffJo

Yes it is, the axiom of infinity is part of the Zermelo–Fraenkel system of axioms:

"Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory."

https://en.wikipedia.org/wiki/List_of_axioms#ZF_(the_Zermelo–Fraenkel_axioms_without_the_axiom_of_choice)

No. What I am saying is that theorems in a field of mathematics need to be based on some set of accepted truths that are called the axioms of that field. Such a set can be demonstrated to be invalid as a set by deriving a contradiction from then, but not by comparing them to other so-called "truths" that you choose to call "self-evident." — JeffJo

You are confused. Perhaps another example will help:

Special relativity is derived from two axioms:

1. The laws of physics are the same in all inertial frames of reference
2. The speed of light in a vacuum is the same for all observers, regardless of the motion

If either of these axioms is ever demonstrated (by experiment or a theoretical argument) not to be true, then all the results of special relativity will be invalidated.

Any 'mathematical truth' is only as good as its axioms - if any of the axioms are false, everything derived from that axiom is invalidated.

The result of that work from a century ago is the Dedekind-Cantor continuum - which is a nonsensical proposition. I'm not sure you read / understood the OP?

Name calling does not win arguments.

If space and time are creations then they must be finite and discrete (impossible to create anything infinitely big or small). But I feel a more direct proof of the discrete nature of space/time/motion is what is required...

A past infinity of time is an impossibility. For example: perpetual motion is impossible, we have motion, hence time must have a start. There are a several other good arguments that the past must be finite that I won't repeat again here. Future time is obviously potentially infinite only. So I don't see you can use past/future time to justify the existence of actual infinity?

I would suggest you instead focus on elapsed time. Is there, for example, an actual infinity of moments in a second? That is a more interesting question. A few thoughts:

- There must be a temporal difference between 'now' and 'then' else 'now' would be 'then'
- The temporal difference can’t be zero / infinitesimal else 'now' would be 'then'
- So there is a finite difference between 'now' and 'then'
- Could that finite difference be infinitely divisible?

A similar question is: 'when you move your hand, do the particles of your hand pass through an actually infinite number of positions?'. Or do they do something similar to a quantum jump of an electron, on a tiny Planck level scale?

Obviously, this is all related to Zeno's paradoxes, which I think are indicative that time and space are discrete. I can't prove it though obviously.

We have been considering models of infinite divisibility here:

https://thephilosophyforum.com/discussion/7320/continua-are-impossible-to-define-mathematically/p1

It seems to me that there are no sound mathematical models of infinite divisibility? So that may lend some weight to the idea that space/time/motion are discrete?

The Difference Between Actual and Potential Infinity

Imagine the real numbers in the interval [0,1]. Is there an actual or potential infinity of them?

Well one answer (the wrong answer) is we can go on dividing forever by 2 (say) so there must be an actual infinity of reals in the interval.

But we can only go on dividing forever in our minds - if we tried this in reality, we'd never finish dividing (process goes on forever - we'd never finish) - so the possibility of infinite division is just a figment of our imagination (like its possible to levitate in your imagination - but not in reality).

There are therefore (according to Aristotle and I agree with him), a potential infinity of real numbers in the interval [0,1]. When we perform division by 2, we actualise the number 1/2. When we perform division by 2 again, we actualise 1/4 and 3/4. And so on. At no point in this process is there ever an actual infinity of real numbers.

So in summary, actual infinity is a purely imaginary concept . It is sometime mentally convenient to regard (say) the set of reals as actually infinite - but that is not telling us anything about reality any more than our ability to imagine a square circle or a fairy.

I understand what an integral is and the concept does not lead to a sound definition of a continuum - unless you can correct me on this?

As you will notice for every point on AB there will always be a unique point on CD i.e. the cardinality of the set of points on AB = cardinality of the set of points on CD. They're both infinite. — TheMadFool

Agreed.

However, notice that a point on AB has a different numerical value to the corresponding point on CD. They are different quantities and so add up to different, not same, lengths. — TheMadFool

That numerical value does not tell us the width of a point and length is the sum of widths of the constituent points / sub-line segments.

Yes, the realist thinks of "past eternity" metaphorically in geometric terms, as an infinitely long line beginning at, say, zero and ending at positive infinity at the "the present", which obviously cannot be constructed — sime

I class myself as a realist but a finite realist, which I consider to be a more 'materialistically real' proposition than a realist who believes in actual infinity - for example of past time:

- If time has a start then it was 00h:00m at the start of time and current time is given by elapsed % 24
- If time has no start then it is UNDEFINED and all current times are also UNDEFINED, so no time

So we cannot make an immediate identification of appearances, memories, thoughts or numbers with points on a physical-history timeline. — sime

So you would hold that even the temporal succession of old thoughts followed by present thoughts followed by new thoughts could be an illusion?

Thoughts are constantly happening so that leads one to have an ever expanding history of thoughts. You hold that even that ever expanding history could be an illusion?

It is hard to argue against such a viewpoint. Denying the evidence of the senses as real is one level of anti-realism. Denying the veracity of our own thought is another level of anti-realism. The second seems impossible to counter logically?

But maybe God is omnipotent within the bounds of common sense / logic? So he can do anything logical but cannot for example, square a circle.

One of the laws of common sense that constrain God's actions could be: 'in order to appreciate a maximum, one has to have experienced a minimum'?

Leading to you have to have experienced pain else you won't get full benefit from heaven.

An axiom is a statement - statements are true or false. End of story.

Treating aleph0 as a mathematical concept leads to paradoxes, eg:

https://en.wikipedia.org/wiki/List_of_paradoxes#Infinity_and_infinitesimals

What is a paradox? It's a contradiction. So a paradox indicates the presence of an reductio ad absurdum argument - IE we have made a false assumption (one of our axioms is false) and it has lead to an absurd conclusion.

In the examples linked above, that false assumption is the axiom of infinity.

The actually infinite is just a mental convenience - it is merely economical mentally to talk about the number of reals in [0,1] as being actually infinite. Of course in reality, it is no such thing - it is an example of potential infinity - those numbers do not take on real existence until we list them (actualise them).

So the axiom of infinity works in our minds, but does not work in any sort of reality - hence all the paradoxes.

Many things work in our minds (square circles, talking trees, levitation) but have no basis in reality. Actual infinity is one such thing.

What were Aristotle's objections to points being zero-dimensional? — TheMadFool

- He felt that something with zero extent like a point could not constitute something with non-zero extent, like a line.
- He also regarded points on a line as a 'potential infinity' in that the points are not 'actualised' until the line is divided. In this way, he justified his believe that 'actual infinity' is impossible.

There maybe others too, I'm no expert on Aristotle.

Are you saying that because there are an infinity of points in any given line that all lines have to be of the same length? — TheMadFool

The mainstream definition of continua (Dedekind-Cantor) does seem to lead to this conclusion:

1. Points have zero dimension
2. A continua has an uncountably infinite number of points
3. All continua have the same structure and cardinality
4. Therefore it follows that all continua have the same length

So point 4 is expressing the fact that:
(size of a point) * (cardinality of the continua) = (size of the continuum in question)
0 * ∞ = 0

Equivalently, if we assume that points have 'infinitesimal' length, then, depending on how you do the math, you get:
1/∞ * ∞ = 0
or
1/∞ * ∞ = 1

And you always calculate the same length for all different lengths of continua.

I'd like to continue the discussion if you don't mind because I see what you mean but I feel, given that mathematicians don't make a fuss about points being zero-dimensional, you're in error. — TheMadFool

Aristotle made a fuss about zero-dimensional points being the components of lines so I feel the question can be regarded as an open philosophical question.

The line AB can be infinitely divided into infinitesimally small non-zero lengths and each length will always have a point associated with it. — TheMadFool

That does not appear to be the case - see the argument in the OP that sums an infinity of infinitesimals to zero (length).

Now you might disagree with the maths in the OP and say an infinity of infinitesimals sums to ∞ * 1/∞ = 1. But this (questionable) maths leads to the conclusion that all continua have the same length - both a segments and its sub-segments are continua so they would both have the length 1.

Then I suppose one could further argue that the infinitesimals on the segment are somehow longer than the infinitesimals on the subsegment - but that's contrary to the definition of continua as all having identical structure (and identical cardinality - uncountably infinite).

The normal definition of axiom:

'a statement or proposition which is regarded as being established, accepted, or self-evidently true.'

The mathematical definition of an axiom:

'a statement or proposition on which an abstractly defined structure is based'

The definition of a mathematical statement:

'A meaningful composition of words which can be considered either true or false'

So an axiom can be false; invaliding any results derived from that axiom.

?

If all you've ever experienced is pleasure and you've never experienced pain then you place less value in pleasure than someone who has experienced pain too.

So we all have to get our fair share of pain on this earth, else we won't appreciate heaven.

The problem of evil is a real show stopper for theologians, they wrap themselves in terrible logical knots trying to circumvent it.

The best I can think of is some form of 'tough love' - that life's harsh experiences actually combine to make us better people in the long run.

I'm more interested in the nature of our reality and the nature of real lines rather than abstract/imaginary mathematical lines. IE is it possible for there to exist in nature something truly continuous (IE of the same nature as our mental models of lines / the real number line). Obviously the question is settled for matter (discrete) but for time and space it is still an open question.

If you assume that the Cantor–Dedekind axiom is true - that the real numbers correspond to points on a line - then we can take the purely mental model we have of real numbers as being infinitely divisible - and apply it to a mathematical line segment - resulting in a mental model of the segment as containing an infinite number of zero-length points (something than does not make physical sense - but we can do it purely in our minds).

But I'm not sure that the Cantor–Dedekind axiom can be said to apply to 'real life' lines - when I wrote the OP, it was buried deep in my consciousness under the category of 'unquestionably the way the world works'. I have since changed my mind about this.

A real life line segment (as opposed to a purely imaginary, mathematical line segment) is something substantial - and it must be composed of something - and the components must have non-zero length - else it is nothing. So I do not see how a physical line segment could ever correspond to our mental model of the real number line (obviously here I assuming some sort of continuous substance in reality to construct the real line segment with - time and space are the only two candidates - no form of continuous matter is known).

Real numbers I think are more like logical/mental labels - they have zero width - they are just a way to label parts of a line rather than the constituents of a real line.

So I think there is probably an equivalence between the real number line and an imaginary/mathematical line segment - both exist in our minds only - so the impossible stuff like them being composed of an infinite number of zero-length points happens in our minds only (where the impossible is possible), but I doubt the equivalence holds to any line with real life existence.

There is obviously also the question of whether time and space qualify as 'something' rather than 'nothing' (Relationism Versus Substantivalism). I am in the 2nd camp on that question.

You're failing to consider the length that corresponds to each point in a line. So, although points are dimensionless, the distance between points have a dimension viz. length.

Considered another way there are an infinite number of points in any given line but the line is constituted of the distances between these points and not the points themselves — TheMadFool

I suppose you can view a line segment as constituted of points or sub-segments. Whichever way though, the length of the constituents has to be non-zero.

Thanks for the links…

If by workable you mean conformity to your private intuition of the continuum, then actual mathematicians have famously wrestled with this. https://plato.stanford.edu/entries/weyl/ — "softwhere

Wrestled with - and consistently failed to achieve - a sound mathematical description of continua - as I also failed to in the OP.

Weyl was not a believer in the ‘Cantor–Dedekind axiom’. He saw the real number as a discrete concept in contrast to the (alleged) continuous nature of time and space:

“The conceptual world of mathematics is so foreign to what the intuitive continuum presents to us that the demand for coincidence between the two must be dismissed as absurd” - Weyl

So he admits that construction of a valid mathematical model of a continuum is an impossibility - and he never achieves such in his work.

The Cantor–Dedekind axiom is highly questionable to my mind. A real number is a purely imaginary concept. It is like a label so it cannot be said to have any width. So an infinite number of real numbers on a finite line segment is acceptable - in our minds only. A line however represents something that can have objective reality. It must be constituted of something - points or sub-line segments, and the parts must have non-zero, non-infinitimsal width - else something becomes nothing.

Weyl was a supporter of the Brouwerian continuum. As I understand it, the Brouwerian continuum has strange attributes - ‘numbers’ in the Brouwerian continuum are allowed to be dynamical, constantly evolving, quantities in that such a ‘number’ does not have a complete, decimal expansion at any point in time - rather it is in a state of constant evolution as its digits grow with time. This means that in the Brouwerian continuum:

- For real numbers a, b either a < b or a = b or a > b does not hold
- The law of excluded middle: for any real numbers a, b, either a = b or a <> b does not hold

I do not class a system with the above two properties as ‘mathematical’ - in the sense that for me, valid mathematics should be built upon the principles of basic arithmetic and logic. I think once these principles are discarded, then we enter the realm of ‘pure maths’ - maths that does not reliably tells us about the world we actually live in - it may tell us interesting stuff about other realities - virtual worlds with different rules to ours - but it does not describe the universe we live in.

So again, we have mathematics failing to come up with a mathematical description of continua.

https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis — "softwhere

I am somewhat discouraged that this system denies the law of excluded middle and by the definition: ‘nilpotent infinitesimals are numbers ε where ε² = 0 is true, but ε = 0 need not be true at the same time’ - this is again contrary to basic arithmetic and logic.

It may well have applications, but a reliable description of the nature of our universe is not one of them.

https://en.wikipedia.org/wiki/Constructive_analysis
https://en.wikipedia.org/wiki/Computable_analysis — "softwhere

"The continuum as a whole was intuitively given to us by intuition; a construction of the continuum, an act which would create by means of the mathematical intuition "all its points" is inconceivable and impossible" - Brouwer

“Space and time are quanta continua… points and instances mere positions… and out of mere positions views as constituents capable of being given prior to and time neither space nor time can be constructed” - Kant

As far as I can tell from your posts, you think that math is some strange form of metaphysics that uses symbols as abbreviations for fuzzy concepts. And then proofs are just fuzzy arguments to be interpreted like mystical literature on the profundities of time, space, matter. — "softwhere

Most of the maths you have linked to falls into the category of contrary to basic logic or arithmetic. Now you might call me closed minded, but maths for me has to obey the principles of basic arithmetic. When advanced maths differs from basic arithmetic I feel it is no longer telling us about the real world - it is describing some virtual reality that is not our reality - so it is therefore not helpful in the pursuit of understanding the nature of our universe.

I have tried and failed using my basic maths skills to construct a description of continua. You claim the logic I use is 'fuzzy' but then do not point out any examples of my ‘fuzzy’ thinking - making me think that you are unable to identify any such - please advise.

My research also indicates that no mathematician has ever come up with a sound mathematical description of a continuum - so I would be interested to learn what your favoured mathematical prescription for a continua is?

Try to focus on the current discussion - my OP - and give me some counter arguments.

This is a philosophy forum, not a mud slinging contest.

As usual, character assassination but no counter arguments.

By definition, ℵ0 is the cardinality of the natural numbers. — quickly

Just a definition. Sort of like defining ∞=∞. Meaningless IMO.

Your argument does not establish that the natural numbers have finite cardinality. — quickly

Why?

My argument in the OP is that alpeh-zero must be finite. What problem(s) do you have with that argument?

If we accept as true that there exists a set that contains the number 1 and, if it contains the number n then it also contains the number n+1? Then we can prove the existence of the cardinalities we name aleph0, alpeh1, aleph2, etc — JeffJo

See the argument given in this OP:

https://thephilosophyforum.com/discussion/7309/whenhow-does-infinity-become-infinite/p1

IE induction leads to the conclusion that aleph0 must be finite.

IMO, the axiom of infinity is nonsensical and leads to absurdities like Galileo's paradox and Hilbert's hotel.

I agree with you, $\sum_{n \to \infty} 1$, tends towards but never attains a purely imaginary concept/state called infinity. (Actual) Infinity is not a number. The arguments given in the OP lend weight to this claim.

Accepting the non-numeric / purely imaginary / unrealisable status of infinity implies:

- The commonly given definition of infinity is wrong: 'a number greater than any assignable quantity or countable number'
- Transfinite arithmetic is a work of pure fiction
- Ideas about space and time that assume the existence of actual infinity are not mathematically sound

Thanks @A Seagull!

This forum has many folk like @tim wood who accept the received 'wisdom' of Cantor and co without question - it is refreshing to talk to someone with an open mind!

BTW Did you know the reason actual infinity is enshrined within maths as a number is that Cantor was a devout Lutheran, believed that God was infinite and believed that God was talking to him telling him to put infinity into maths!

Fast forward to today and ironically, it is mainly the atheists who believe in infinity - theists and deists often question its existence and rightly so.

My interest in maths stops when maths stops telling me about the nature of reality.

Transfinite maths and reality are greatly in opposition. ∞+1=∞ tells us that there exists something, that when it is changed, it does not change. This is not the reality I live in. I doubt that it is even possible to define a viable virtual reality that supports the 'features' of transfinite maths - how can something change and not change at the same time?

I believe that for example, non-euclidean geometry maybe telling us something about our reality and it is possible to build logical virtual realities on that basis. But infinite set theory... no.

Does infinity have existence beyond our minds? All sorts of things like fairies, square circles, can exist in our minds but only a subset can take a concrete form in reality.

I believe that an actually infinite set or some substance that extends 'forever' cannot exist in reality. We can only imagine such things.

Thats not a counter argument.

Did we define "exist"? Seems like it might be a good idea. — tim wood

OK give me an example of an actually infinite set from nature.

I am, about some things, like 2+2=4. But for you I'll reopen a very open-minded offer I made to you earlier. I give you one dollar bills, and for each one you give me a five dollar bill. Can't get much more open-minded than that, or is that too much for you? — tim wood

Arithmetic is defined, actual infinity has no sound definition. So its fair game for the open-minded.

You seem to have a childish obsession with point scoring through cryptic remarks.

All your counter arguments have been countered so you resort to mud slinging. Just typical of you.