Yes of course but how can you actually measure (to compare with calculated) a probability in experiments in a way that proves anything like no its not continuous actually since you will always have finite number of measurements. It may be possible to see patterns if you did trillions of experiments but frankly who has done that under perfect identical conditions to see any difference?

The theory works well by assuming the continuous calculus used to derive everything. But it would work ok with limits too (albeit harder).


The real theory is not one that assumes spacetime is a grid and calculus is its approximation. No the real theory the way i see it is more complicated than that and its limit is the continuum theory the way we see spacetime currently etc.

What i am searching is examples where what was produced cannot be seen as a sequence of limits also getting us there.

I am really trying to see if there is a true problem in math between finitism https://en.wikipedia.org/wiki/Finitism and the rest. Why cant you see things as finitism but without necessarily specifying where it ends...Wont the unspecified end limits recover everything anyway?

I mean i can see the differences in cardinality of sets. I can see how 2^(1/2) takes infinite number of natural numbers and the real line is infinitely denser than the rationals etc.

I suspect that some of the pathologies in physics come from taking the infinity literally in many things though but this is a different problem one that limits wont solve only reveal where it starts failing.

I am just asking for some result that its impossible to obtain if you see infinity as only a limit with finite mathematics.

Finite mathematics should still be able to see the same theorems of calculus for example. If you stop the limit you have a suitable approximation of the continued thing say. True reframing of physics though wont require to stop somewhere but to reproduce the continuum as limit of something else.

There is something else that gives you a behavior that is satisfactory to describe assuming continuum up to a point. In a similar manner to Newtonian physics being the low speed limit of Relativity or classical physics the limit of Planck's constant going to 0 the real theory has the current theory framed inside a spacetime where you perform calculations as if you can have infinite precision (eg action integrals etc). I suspect this is only an approximation. In other worlds the infinity itself is the problem when taken literally in certain key questions but not in the vast majority of things.

However what i was asking is if there is any significant across mathematics result that is only possible by realized infinity and not endless systems of limits. (i mean its harder to do less compact but still the same thing not?)

I think the answer is no, but by no means should you trust me on this.

(I was just reading this topic by the way, which might be interesting:

http://math.stackexchange.com/questi...-mathematician ,

I have pretty much no idea about almost anything that is being discussed there (well, I very vaguely understand some stuff; in math we do have to take a few courses in functional analysis, differential geometry, differential equations and whatnot, all of which are important for physicists; not particularly my favorite subjects though) but it does not seem to me that there are any clear applications of logic/set theory to anything physicists do; I suppose I would expect something like this would be mentioned in a topic like this if it did).

Even if one accepts the Axiom of Infinity, it might still be argued that this algebraic definition fails to capture the essence of what integers are.

I'm fairly sure that a topological definition is possible, by considering a discrete linear order with no top or bottom element. Once an arbitrary element of this order has been chosen to be zero, addition and multiplication can then be defined canonically, in terms of the order.

This corresponds more to how integer arithmetic is taught to children. First the order is introduced (counting), and then addition is defined using the order (e.g. 2+3 is the result of counting 3 forwards from 2).

https://en.wikipedia.org/wiki/Total_order

On accepting the Axiom of Infinity:

http://math.stackexchange.com/questi...om-of-infinity

In particular:

(1) Nowadays, generally, one doesn't accept or reject (ie, ""accepting" and "rejecting" isn't the right notion to refer to) things such as Axiom of Infinity, Axiom of Choice, Continuum Hypothesis and so on, specially if one works in foundations. What people in these foundational fields do is to study what happens in MANY of these different axiomatic set theoretical systems. It is a common thing in set theory to study questions such as: What happens if we assume ZFC (the standard system of axioms)? What if we assume just ZF (the standard system without Axiom of Choice)? How about ZFC+CH (the standard system with the Continuum Hypothesis added as an axiom) or ZFC+(-CH) (the standard system with the negation of the Continuum Hypothesis added as an axiom)? And so on.

(2) As is mentioned in the topic, if one does not assume the Axiom of Infinity, then our universe of sets becomes fairly poor: if we can't collect a "countable amount" of sets into a single set (so that we may suitably define one such a set as what we may want to call the "set of the natural numbers"), we will end up with a countable universe and, say, it won't be possible to construct the real numbers.

What one needs to understand is that nowadays mathematicians don't generally make a choice for a canonical foundational system (which is currently ZFC), based on philosophical or religious beliefs. They just pick a system which they feel works the best and gives us lots of objects to work with. We WANT to turn the naturals into a set, and use this set to construct biggers and biggers sets. It makes mathematics much more interesting.

Anyway, I don't really understand the rest of your comment.

What do you mean by "this" "algebraic definition" or by "topological definition" of the integers? I didn't give any definitions for the integers anywhere in this thread (or anything at all, in fact; well, except for the integers mod 2). It is not clear to me what an "algebraic" definition is, and what a "topological" one is.

A definition of the integers means a (formal) process of construction for them from the axiomatic system you are working with (which will most likely be ZFC); the most common construction of the integers is as a set of equivalence classes of natural numbers (under a suitable equivalence relation) with a suitably defined addition, multiplication and order; here of course we are USING the fact that we already have the naturals. So, previous to the process of defining the integers, one needs to define the naturals; here is when one makes direct reference to the Axiom of Infinity; being a bit vague, you define the set of natural numbers as the set which is the intersection over the class of sets which contains the empty set and are closed under the successor function (the fact that such sets exists is EXACTLY what the Axiom of Infinity is).

One may refer to A system of integers as (for instance) an ordered commutative ring with well ordering principle for its positives. In case this what you mean by "algebraic definition", well, it's not exactly "algebraic", we DO have to talk about order here, and it is certainly not a definition: one needs to prove existence and uniqueness of such a supposed "ordered commutative ring with well ordering principle for its positives". Existence means exactly doing a construction process as described in the above paragraph (because such a system could have failed to exist; say if you had named "A lastcardcharlie system" to be a set such that 0,1 are elements, 0=1 and 0≠1, well, you can see the last two requirements are in contradiction with one another; no thing will satisfy this) and uniqueness means proving that any such two systems are isomorphic; say, there are many different groups, rings, fields, vector spaces, modules, algebras, metric spaces, topological spaces and so on; they are not all isomorphic within themselves (say, the rationals and the complexs are non-isomorphic fields) so one wouldn't talk about "the field" or "the topological space", but (as soon as we prove it) we can talk about THE ordered commutative ring with well ordering principle for its positives, ie, THE integers.

I (mis-)took your comment I quoted in my previous post as intended to give defining properties of the integers.

As for the "algebraic" part of your question, I think you answered that yourself later on in your post.

As for the "topological" part, the order on the integers is topological in the sense that it is the specialization order of the Alexandroff topology of upper sets, i.e. the order could be replaced by this topology without any loss of structure.

https://en.wikipedia.org/wiki/Alexandrov_topology

Awesome. To respond

"Even if one accepts the Axiom of Infinity, it might still be argued that this algebraic definition fails to capture the essence of what integers are.",

then, I would say that being a ordered comutative ring with the well ordered principle for the positives does indeed characterizes completely the integers: there exists such a system (which one can show by the above mentioned construction) and it is unique up to isomorphism (ie, between any two such systems, there exists a bijection which preserves addition, multiplication and order, and so, in particular, will preserve any topology which is somehow induced by its order).