20

Here is Cantor in his own words (from his influential 1887 letter to Weierstrass): "I begin from the supposition of a linear magnitude ζ which is so small that its product by n , ζ · n, for every finite whole number n however great is smaller than unity, and then prove, from the concept of a linear magnitude and with the help of certain propositions from ...


17

They are two different approaches towards understanding the notion of the infinite. (They can be used for finite quantities as well, but they coincide in that case, so it's a bit boring.) There are other approaches as well, but they tend to be derivative concepts of at least one of cardinality or ordinality. These two concepts are different when it comes ...


17

It's no solution to postulate a primordial source as a remedy against infinite regress. The concept of a primordial source prompts at once the question for its cause. To say it is "causa sui" - the answer of Christian philosophy - does not answer the question but rejects it. My conclusion: We must not overestimate the power of pure reasoning. Instead, we ...


15

We must first distinguish between what is physically possible — what it is possible to actually occur — and what is imaginable, or logically possible under certain premisses. Remarks about the logically possible Initial approaches Logic itself — which I will take to mean classical propositional logic — has very little to say about ...


14

Disbelieving in infinity is going to cause you problems only if you are a mathematical realist, meaning that you believe that a number like 5 has some independent ontological existence that infinity lacks. In the case that you think that both 5 and infinity are just useful concepts then the problem disappears. There are plenty of concepts we find useful in ...


14

Most answers are misinterpreting your question. Whether it be space-time itself, the multi-verse, or the Flying Spaghetti Monster you would like to know if something had to first exist for infinity due to the problems with infinite regression. If we assume the Big Bang Theory is correct, then there is no "before" the Big Bang in the usual sense of the word ...


13

This is very similar to Thomas Aquinas' second 'way'. Aquinas was a 13th century catholic philosopher and wrote Quinque viæ, five ways to God. The second is The Argument of the First Cause. The five ways are controversial and very debatable. Richard Dawkins famously criticised them in The God Delusion. He claims it is a form of special pleading to say that ...


13

The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but rather that any extension of "size" to infinite sets will be paradoxical, it has to discard one ...


11

Terminology changed somewhat, and much of what used to be called "logic" as late as early 20th century is now called epistemology, for more details see What are the differences between philosophies presupposing one Logic versus many logics? Posterior Analytics covers mostly that epistemological part of logic. What Aristotle describes is what later was coined ...


10

The concept of infinitesimal small and infinitely large numbers has been been formalized by the mathematical domain of non-standard analysis. The field of rationals (QQ,+,*) embedds into the ring (Omega_QQ,+,*). Elements of the latter are the equivalence classes of sequences of rational numbers; two sequences are considered equivalent when their ...


10

Aristotle said the past is infinite because, for any past time we can imagine an earlier one. Aristotle's arguments aside, this is what people mean when they speak of an infinite past: for any time x, there exists another time y such that y precedes x. Colloquially, "there is no first moment in time". If time has a beginning, it means that there is a time x, ...


9

Infinities in the sense of Cantor are cardinalities. There is no cardinal equivalent to infinitesimals. If you are counting, the smallest thing other than zero is one. I do not know Cantor's own argument on this account, so I have to skip over your first two questions. But modern mathematicians do not see the two as in any way related, with a few ...


8

There are some people who believe our universe is contained within a "multiverse" which contains all possibilities (which personally I find a depressing prospect, since it would arguably reduce to meaninglessness any given event happening anywhere). The "multiverse" is highly speculative, however, and there are plenty of other people who disbelieve in it ...


8

Most physicists don't accept infinities for a very obvious reason: such infinite physical objects are not quantifiable! That is, we can't measure them or even prove that they are infinite. Through the history of physics, infinities were raised in formulas, and usually in these cases the formulas were thrown away, considered as incomplete, or they kept ...


7

1) I see it as infinite, never-ending process. YES, It is : every infinite process is never-ending. 2) I do not see what you get in result. I do not see infinity-hotel accomodating 2 infinities of individuals. With an infinite Hotel it happens ... Two "infinite" set (if we assume that here infinite stay for countable) "added together" gave us a new "...


7

It seems to me there is a fundamental contradiction between two parts of your question. First you say: "I am an ultrafinitist". Then you ask how you should interpret modern mathematics. But ultrafinitism is an interpretation of mathematics. So either you subscribe to that interpretation, in which case you surely have no need to ask what it is, or you do ...


7

OK, I'm going to have a go at this. An argument can go 3 ways: A circular path. Infinite regress. Hmm... let's just call it stop condition for now. Circular arguments These are outright nonsense. A thing cannot prove itself. Oil floats on water because oil floats on water. Nonsense. The premises are right, but the argument is nonsense. Compare with: ...


6

Actually, it's the opposite. Here's a simplistic overview of why: Suppose you do a finite amount of mathematics (prove a finite number of theorems from a finite axiom set). But by the incompleteness theorems, there are some theorems that simply cannot be proved. Thus, no finite amount of mathematics suffices to encompass all of mathematics. It's instructive ...


6

V-omega, the set of hereditarily finite sets is a model of the theory you get by taking ZFC and replacing the axiom of infinity with its negation, and it is bi-interpretable with Peano Arithmetic (so indeed, Gödel's incompleteness theorem still applies). For more, see e.g. https://math.stackexchange.com/questions/107639/what-are-the-consequences-if-axiom-of-...


6

How you deal with infinity depends on what your priorities are. If you care only about sheer cardinality — as Frege did, as he was considering set theory — you can quite easily have an infinite set, with a proper subset having the same size. But to have this, you must ignore most if not all structure in the infinite set, and define "size" by considering ...


6

Your question trades on a mistake. Truth and falsity are properties of sentences. The law of the excluded middle therefore is a rule about sentences. An undefined operation isn't a sentence hence it is neither true or false, but this doesn't imply that it has some kind of third truth value. It simply lacks a truth value all together. My toaster also lacks a ...


6

Here is the short mathematical answer : An ordinal is compact if and only if it is a successor ordinal. A cardinal is compact if and only if it is finite. Here we are assuming the natural "order topology" is used. The result for cardinals follows from the result for ordinals since an infinite cardinal must be a limit ordinal. If λ is a limit ...


6

The question you are asking had a consensus answer that agrees with yours until the end of 19th century. All infinity is like "continuous generation of numbers", or what philosophers called potential infinity, there is neither physical actual infinity that could result from completing such a process, nor even mathematical one. This view was expressed most ...


6

Is the axiom of infinity truly an axiom? Yes, it is an axiom of set theory. But in mathematics an axiom of a theory does not have to be plausible according to our everyday intuition. The only requirement it has to satisfy: The axiom does not contradict the other axioms of the theory. Of course axioms should not to be chosen arbitrarily. They should serve ...


6

If you reject all definitions of what we would use the word "infinity" to mean, then it's tautological that "infinity" cannot be defined. Similarly for the word "infinite".


5

Kronecker studied philosophy in his youth, in particular the doctrines of Hegel. I think Hegel identified the absolute giest (mind/spirit), an actual infinite reached the final resolution of his dialectic, as God. Hegels doctrines were wildly popular. He had declared his faith in a 'concrete God'. I assume his phenomenology of spirit was to construct him, ...


5

However, this supposes that the structure of reality should be explainable or understandable, for which I see no a priori reason. That's the problem, or rather, the reason why infinite regress isn't a problem to you. Infinite regress doesn't explain anything, which is why it is a problem for people looking for explanations. Note that this problem isn't ...


5

Let Γ be the class of all impossible sentences, i.e., Γ = {φ : ¬◊φ}. Someone claiming that nothing it impossible is simply claiming that Γ = ∅. That commits them to the thesis that no formula is necessarily false, not that "the impossible is impossible" (whatever that means). The thesis is obviously false (i.e. Γ ...


5

It seems to me that your question is ill-posed. With that I mean that the question is not well defined: take for example the classic problem "If a tree falls but no one is there, does it make any sound?" The problem with this is the unclear definition of "sound", so if one does not define it better, the question is ill-posed. Back to your problem. ...


5

It is simply not the case that "anything must happen" in an infinite sample space. Consider the universe consisting of three possible states, A, B, and C. The universe recurs endlessly forever, taking states A, B, A, B, A, B, ... Even though there are infinitely many instantiations of the universe, C never occurs. The best you can say is that some state ...


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