If you rule out infinite sets you have to admit that measures of geometrical objects are not sets. For example:
- how many different lengths can have a segment?
- how many different directions there are? ( intended as different unit vectors in 3d space )

I think ancient Greek mathematicians had this point of view: there are discrete numbers used for counting, that are objects with zero dimension; there are lengths, with dimension 1, areas with dimension 2, and volumes with dimension 3. And each of these "kinds" of measures is not comparable with the others. So, the question of how many "points" are needed to form a line does not make really sense, since objects of dimension 0 and 1 are not comparable. So, no need of infinite sets.

There is an infinite possibility of line segments with different lengths but if we were to join them, we would never complete the task(if we add ever increasing lengths), an infinite convergent series sum is a different mathematical object compared to an infinite divergent series as we can have well defined results for earlier one.Some philosopher use the term potential infinity which always refers to a finite quantity in any instant of time therefore the term is misleading.I think sometimes a geometrical object does not correspond to the arithmetical results we obtain.
I Think the case with vectors in 3d is the same,we can say the norm of the vector containing the sum of all unit vectors in different directions (unlimited ) will be zero, but if we can argue consider the collections of and infinite amount of vectors in opposite direction to each other, however the resultant norm cannot be determined.Similar case arises in series.
The measure of Cantor set is zero and that requires the existence of an infinite set.
Can you clarify on the representations of geometrical objects using an infinite set, I think finite sets suffice.
The behaviour of uncountable infinite set gives bizzare results as mentioned above.
If we adopt a non platonic view of mathematics, and on basis of that, we cannot justify infinity.Infinity can be idealized but we can never point to it in the universe.
Hilberts paradise I guess welcomes it.
Pardon my ignorance.

an infinite convergent series sum is a different mathematical object compared to an infinite divergent series as we can have well defined results for earlier one — Wittgenstein

In every case the series (convergent or not) is made of a countable set of 1-dimensional segments of non zero measure. You add segments to obtain a segment, not points.

The measure of Cantor set is zero and that requires the existence of an infinite set.
Can you clarify on the representations of geometrical objects using an infinite set, I think finite sets suffice. — Wittgenstein

If you think of a segment as made of a set of points, it must be an uncountable set of points because a countable set of points would have zero as 1-dimensional measure.
If you instead say that points cannot be used to build a segment because points have no measure, then you agree with me that points and segments are two different kinds of "objects", so one cannot be built starting from the other

In every case the series (convergent or not) is made of a countable set of 1-dimensional segments of non zero measure. You add segments to obtain a segment, not points.

~~Mephist~~
I agree that we add segments to obtain a segment, however l have two questions :
1.If the segment is not made up of points, is it a non zero measure or something else ? ( I tend toward your view )
2.Can a divergent series be obtained from an uncountable set, ( I think it can be ) but can a convergent series be obtained from a uncountable set ? ( A sum that is definite must have a fundamental difference to a divergent one )

I think it is reasonable to take segments and point as different cause we can remove the paradox of infinite points or the use of uncountable set of points.

There is an infinite possibility of line segments with different lengths but if we were to join them, we would never complete the task — Wittgenstein

Having already admitted the existence of uncountably many line segments, you can take their union in one application of the axiom of union to join them. It's hard for me to understand admitting the existence of a collection of sets but denying their union. Of all the axioms of set theory that are commonly debated or questioned (choice, powerset, infinity, foundation), union is not one I've ever heard anyone question.

1.If the segment is not made up of points, is it a non zero measure or something else ? — Wittgenstein

A segment (1-dimensional) has a non zero 1-dimensional measure, and can be only made up using objects with non-zero 1-dimensional measure. ( I am speaking about the view of mathematics that ancient Greek mathematicians - for example Archimedes - had )

2.Can a divergent series be obtained from an uncountable set, ( I think it can be ) but can a convergent series be obtained from a uncountable set ? ( A sum that is definite must have a fundamental difference to a divergent one ) — Wittgenstein

If you believe in finitism (as you proposed) there are no infinite sets, neither countable nor uncountable.
Again, this is Archimedes' philosophy of mathematics, not mine.
However, a series is BY DEFINITION a sum of terms indexed by integers. An integral on real numbers instead is a sum of terms indexed by real numbers, that are an uncountable set in ZF Set theory. Then, in modern mathematics a convergent sum of an uncountable set of terms is perfectly normal.

Finitism overlooks the fact that the practical use of 'infinity' is merely to denote an unspecified unbounded number. Consequently, If infinity wasn't part of mathematics, then mathematics could not be used by science to describe current physical laws, rather it would only document previously verified event occurrences.

In C programming, the equivalent symbol to infinity is the volatile keyword. When a C program declares a volatile object, say "volatile int myInteger" , the program is declaring that the value of "myInteger" isn't specified by either the programmer or the program itself, but that the value will be supplied later at an unspecified time by the environment. The specification of an 'infinite' number of iterations in a computer program, say while(true) {..}, is therefore equivalent to writing while(volatile int myInteger){....}.

Unfortunately, set theory and logic do not possess an equivalent concept. The closest axioms they possess is the Axiom of Choice, but this axiom is flawed because it is considered to be either accepted or rejected universally across all sets, and it also fails to discriminate sets which are volatile and bounded from sets which are volatile and unbounded. Consequently the status of the axiom of choice remains confusing and controversial.

In my opinion, the historical cause of debates over the existence of infinity is the result of logicians failing to recognise that the semantics of logic and maths isn't fully a priori.

In C programming, the equivalent symbol to infinity is the volatile keyword. — sime

Jeez that's not true. A volatile variable is one that is, for example, mapped to an external data source. Declaring a variable volatile tells the compiler that it can't depend on nearby code statements in order to optimize the variable.

This simply has nothing at all to do with transfinite ordinals and cardinals as understood in math. It's apples and spark plugs.

it also fails to discriminate sets which are volatile and bounded from sets which are volatile and unbounded. — sime

Those terms have no referents in math. I am not sure where you are getting these notions. Your ideas about the axiom of choice and mathematical infinity are idiosyncratic to say the least.

In C programming, the equivalent symbol to infinity is the volatile keyword.
— sime

Jeez that's not true. A volatile variable is one that is, for example, mapped to an external data source. Declaring a variable volatile tells the compiler that it can't depend on nearby code statements in order to optimize the variable. — fishfry

sorry, I should have said infinity is equivalent to volatile and unbounded. I am saying volatile and unbounded is equivalent to the specification of an infinite set, considered as extension, in cases where the infinite set is not directly defined in terms of a constructive algorithm.

This simply has nothing at all to do with transfinite ordinals and cardinals as understood in math. It's apples and spark plugs.

it also fails to discriminate sets which are volatile and bounded from sets which are volatile and unbounded.
— sime

This has no referent in math. I am not sure where you are getting these notions. — fishfry

Likewise, Transfinite ordinals divide into those which are specified constructively as tree-growing algorithms and those which denote unspecified trees to be supplied by the environment, whether bounded or unbounded.

Transfinite ordinals divide into those which are specified constructively as tree-growing algorithms and those which denote unspecified trees to be supplied by the environment, whether bounded or unbounded. — sime

You have a link in support of your ideas? They are very strange. I don't want to flat out say they're wrong, since my ignorance is vast. But I know a little about ordinals and I can't correspond your words to anything I know.

infinity is equivalent to volatile and unbounded. — sime

I don't know what is a volatile and unbounded set. Can you provide some examples so I can understand what you're saying?

Volatile is not a term of art in math at all. And its use in C programming is very specific as I think we agree. It just tells the compiler not to optimize the variable.

The integers are unbounded because you can't draw a finite circle around them all. The unit interval is bounded since all its elements are within 1 unit of each other. Yet the unit interval has far larger cardinality than the integers. So I am not sure what you're trying to get at.

Regarding ordinals, you say that there are some which "denote unspecified trees to be supplied by the environment ..." That's .... well again it's idiosyncratic. There are ordinals which are computable and ordinals which are not. Are you thinking of the Church-Kleene ordinal or one of the other exotic countable ordinals?

I really don't know what you mean by saying (some) ordinals are supplied by the environment. That's not what ordinals are. Ordinals are order types of well-ordered sets. They don't take on values like memory locations in an executing computer program. I think perhaps you might be trying to push programming analogies farther than they can go.

I don't know what is a volatile and unbounded set. Can you provide some examples so I can understand what you're saying? — fishfry

Yes, any programmer's use of an infinite FOR loop. We all know in practice that infinite loops are, in a pragmatic sense, merely finite loops whose termination condition isn't specified by the program. In other words, the termination of the algorithm isn't internally constructive from the point of view of the program itself.

Volatile is not a term of art in math at all. And its use in C programming is very specific as I think we agree. It just tells the compiler not to optimize the variable. — fishfry

Right, i'm am not so much referring to compiler mechanics, as to the logic of volatile types. Programmers use a richer notion of logic than is used by traditional set theory that equivocates internally constructed sets with externally supplied sets. The consequence of this is mistake is the kludge known as the Axiom of choice that allows the specification of arbitrary unbounded sets, but only for unbounded sets, effectively conflating arbitrariness with unboundedness [/quote]

The integers are unbounded because you can't draw a finite circle around them all. The unit interval is bounded since all its elements are within 1 unit of each other. Yet the unit interval has far larger cardinality than the integers. So I am not sure what you're trying to get at. — fishfry

Whenever we refer to an integer, we are either referring to a integer which we ourselves have or will construct using an algorithm in our possession, or we are referring to an arbitrary integer that is to be delivered to us by some external source. Constructivists make the mistake of conflating existential quantification with construction. It is a mistake, because, say, we cannot run society on software that uses only predictably terminating bounded for loops. Platonists on the other hand, while rightly insisting that non-constructed sets are indispensable in practice, wrongly locate the source of that indispensibility to a priori notions of existence.

$\sum_{n=1}^{n=infinity}$ $\frac{-1^n}{n}$ , as you can see in this series we have not indexed the set using negative numbers, and l think the series will not be well defined if we do not restrict R to natural numbers.( countable )

If you want to reconcile geometrical objects with only finite sets, this approach has been used.To quote from the article mentioned below,

distinction between a field such as R and a Galois field. In the latter, given the multiplicative neutral element 1, there is a prime number p such that p⋅1=0. p is called the characteristic of the field. It can be shown that if p is the characteristic of a field, then it must have pn elements, for some natural number n. In addition Galois fields are the only finite fields

Further more if the galois field has characteristic 3, then 2+1=0.
There is a problem in this theory, in that x=-y cannot be thought of as a straight line but ( it is ).
I hope we can discuss this article, to make arguements move forward in the thread.

https://plato.stanford.edu/entries/geometry-finitism/supplement.html

I am unfamiliar with C programming and l would feel comfortable to relate a similar problem in applied mathematics, sometimes in mathematical modeling, where the constraints even as small as the number 7 can be treated as infinity as from 7 and onwards, the program gives the maximum output.I think the axiom of choice is similar to the Euclids parallel axiom, they should be used or ommited depending on the situation.
Btw, I have a question regarding algorithm, I read somewhere Von Nueman saying that it is impossible to have an algorithm that can strictly generate random numbers.Is there even such a thing as a strictly random number.

OK, but I don't understand what's your point here..

Yes, I didn't finish to read the article in detail, but it sounds very interesting.
However, I am not surprised that there is an axiomatization of euclidean geometry that doesn't make use of infinite sets. What I said is a very simple and obvious thing: if you don't allow the existence of infinite sets, you have to treat segments as a different kind of thing than a set of points.

if you don't allow the existence of infinite sets, you have to treat segments as a different kind of thing than a set of points.

But in the galois field, they treat the segment made of points but don't use infinite sets, the one mentioned in the article.Can you send me any article,book recommendations that make your position clear to me cause l may be confusing your point here, I hope not.

OK, but I don't understand what's your point here.

I am terrible at explaining things but at same time I am wondering which one is that which you dont understand.
Can you quote it.

∑n=infinityn=1∑n=1n=infinity −1nn−1nn , as you can see in this series we have not indexed the set using negative numbers, and l think the series will not be well defined if we do not restrict R to natural numbers.( countable ) — Wittgenstein

I agree that this series should be defined on natural numbers, but I don't understand to what part of my post is this related. I said that a series, by definition, is a sum of a countable set of elements. Is this an objection to this point?

Then, in modern mathematics a convergent sum of an uncountable set of terms is perfectly normal.

I wasn't objecting to a countable set, but to an uncountable set.
Nevertheless both are infinite sets.

Well, I reviewed the definition of Riemann integral and I see that it is defined as a series (sum over set of intervals). So yes, I was wrong: there are no sums defined on an uncountable set of terms (for what I know).

Can you consider this arguement against an infinite set,
What is the probability of an event happening over an infinite amount of events, it would be zero.We can go on and prove that the possibility of any event happening will be zero but that would be absurd if we applied it to the world.

Can you consider this arguement against an infinite set,
What is the probability of an event happening over an infinite amount of events, it would be zero.We can go on and prove that the possibility of any event happening will be zero but that would be absurd if we applied it to the world. — Wittgenstein

Are you arguing against infinity in math? Or just in the world? It's perfectly clear that we all have a intuition of the natural numbers 0, 1, 2, 3, 4, 5, ... They are generated by the simple rules that:

* 0 is a number;
* If n is a number, so is n + 1.

Put those rules into a Turing machine and they crank out the endless sequence of natural numbers.

Nobody claims that the infinite collection [not yet a set, that requires the axiom of infinity] of natural numbers is instantiated in the natural world. It only exists as a mental abstraction, like justice or traffic laws or Captain Ahab.

Or are you arguing that you accept mathematical abstractions but denying that they're physically real? That's perfectly sensible.

But by referring to the world, you are out of the realm of the abstract and instead making a trivial point about the world. That weakens your argument considerably.

In any event, infinitary probability theory is well understood and allows for probability zero events that nevertheless may happen. For example the probability of picking a random real number and having it be rational is zero; yet the rationals are plentiful. [That's not a precise statement, but it can be made precise without loss of intuition].

How will you generate negative number using n+1 ?
( natural numbers are infinite nevertheless)

Nobody claims that the infinite collection [not yet a set, that requires the axiom of infinity] of natural numbers is instantiated in the natural world. It only exists as a mental abstraction, like justice or traffic laws or Captain Ahab.

My problems is with the use of infinity as a number in certain mathematical problems, for example the lim 1/x as x approaches 0 will be written equal to infinity.But using an equal sign with infinity can be challenged even in its abstract form, l do understand the theory behind limits but in certain cases referring to infinity, mathematicians treat it as a number, not a concept.
Let's suppose in an ideal world, it is treated merely as a concept, Quine would argue that there are certain infinities allowed which find applications in science but those which are not applied to natural world should not be given equal weighting to the earlier ones.

Or are you arguing that you accept mathematical abstractions but denying that they're physically real? That's perfectly sensible

Consider a mathematical abstraction which describes the world ( quantum mechanics for eg ), l think some mathematical abstraction can co-exist with the real world although some don't.If such an abstraction does not agree with reality as we know, we can drop them even if they are consistent mathematically.
On the other hand, l am not talking about their existence as cats or dogs exist ( that would be stupid ) but their metaphysical existence.Earlier on you mentioned generating an unending natural numbers, but can you ever list them, l am not talking about physical limitation but the nature of infinity would not allow even the fastest computer to list them all, hence it can be contended that infinity as an abstract mathematical can never be produced theoretically, hence it does not exist.

In any event, infinitary probability theory is well understood and allows for probability zero events that nevertheless may happen. For example the probability of picking a random real number and having it be rational is zero; yet the rationals are plentiful. [That's not a precise statement, but it can be made precise without loss of intuition].

I used the probability theory as an example because it is related to the world, but since you claimed it can be made intuitive, l would like you to clear that up. If someone were to talk of negative probability ( fenyman did l think ), where we consider things we do not observe but which do occur in the real world.( l can be wrong here ), that is more understandable than the use of infinity in probability theory.There is also another problem, if all the probability are 0, then the the total probability of all events will not give 1.That is against the law of probability.Further more if you take natural numbers as the domain of probability distribution, it would be not be well defined.

. It's hard for me to understand admitting the existence of a collection of sets but denying their union

I would like to quote this for explaining my point of view regarding your objection.

Since a mathematical set is a finite extension, we cannot meaningfully quantify over an infinite mathematical domain, simply because there is no such thing as an infinite mathematical domain (i.e., totality, set), and, derivatively, no such things as infinite conjunctions or disjunctions

infinity is clearly not a number — Wittgenstein

The number is one.

Probability should be considered as part of set theory, rather than classical set theory being considered to be a foundation for probability theory. For the semantics of probability is the semantics of empiricism and directly concerns both "volatile", uncertain and undetermined empirical sets as well as logically constructed sets, yet unfortunately these two meanings of probability are obscured if probability theory is reduced to classical set theory.

For instance, consider a sigma algebra (i.e. a sample space) denoting the set of possible outcomes for an infinite sequence of coin tosses t(1),t(2),... i.e. a coin toss process whose length is undefined a priori. Coin tosses aren't a mathematical concept but an empirical affair, and conversely mathematics isn't an empirical theory. Therefore it makes no sense to insist that the sigma algebra of infinite coin tosses must be constructive. For it might well be the case that a sequence of coin-tosses is truly random in the sense that cannot be represented by any computable function. This is the case if it is believed that for any computable binary function f there exists a subsequence of observations t(1)..t(n) that isn't equal to f(1)...f(n). Unfortunately, set theory fails to distinguish externally observed processes from internally constructed processes, hence the reason why finitists and infinitists continue to argue past one another.

" Unfortunately, set theory fails to distinguish externally observed processes from internally constructed processes, hence the reason why finitists and infinitists continue to argue past one another. "
"Well consider For it might well be the case that a sequence of coin-tosses is truly random in the sense that cannot be represented by any computable function "

I agree with the first paragraph, it is a great observation in my opinion.However for the second paragraph, can you consider two guys that have unending supply of coins, and each toss say for n times, we can calculate the probability for the next test by n/2^n.Well l dont think a coin toss can be that random, however if we were to pick a prime numbers out of real numbers, l think we would not have a comparable function for it.

Therefore it makes no sense to insist that the sigma algebra of infinite coin tosses must be constructive.

Since in the end you mentioned the defect in set theory, l think we can argue for a constructive case, where a statement is either true or false.
My knowledge in this field is weak and it would be of great help if you can provide some resources on treatment of probability theory from a non classical point of view.

I couldn't get what you were saying, can you elaborate.

Infinity amounts to one.
You take one and you start stretching it out; that's infinity.