31 votes
Accepted

If the universe is finite does that nullify Godel's incompleteness, halting problem, and Church-Turing thesis?

First of all, this question presupposes that mathematics is limited to describing the physical universe. Even as finite beings in a finite universe we can still try to reason about hypothetical ...
user avatar
20 votes
Accepted

What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal?

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 ...
user avatar
  • 40.8k
17 votes
Accepted

Is there an alternative to Cantor's cardinalities that makes proper subsets smaller than their sets?

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 ...
user avatar
  • 40.8k
16 votes

Does the impossibility of an infinite regress prove God exists?

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 ...
user avatar
  • 20.6k
16 votes
Accepted

Does the impossibility of an infinite regress prove God exists?

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 ...
user avatar
15 votes
Accepted

If we live in a simulated world, doesn't there have to be a first world that's real?

We can not carry the argument past the first step because if our physical laws are simulated then we know nothing about the "physics" of the world that does the simulating. In particular, it may make ...
user avatar
  • 40.8k
13 votes

Infinite past with a beginning?

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, ...
user avatar
  • 3,759
12 votes
Accepted

How should one interpret modern mathematics if one doesn't believe in infinity?

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 ...
user avatar
12 votes
Accepted

Do all epistemologies suffer from the "regress of justifications" problem?

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 ...
user avatar
  • 40.8k
11 votes

What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal?

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 (...
user avatar
  • 20.6k
9 votes

Does the impossibility of an infinite regress prove God exists?

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. ...
user avatar
  • 211
9 votes

If the universe is finite does that nullify Godel's incompleteness, halting problem, and Church-Turing thesis?

The halting problem doesn't go away, even in the modified variant that would exist in a finite universe. A modified halting problem that instead of "Does this ever halt?" asks "Does ...
user avatar
  • 1,193
8 votes

Are infinities in physics (or in any other materalist philosophy) actually possible?

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 ...
user avatar
  • 325
8 votes

Infinity and the universe

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 ...
user avatar
8 votes

Does science require the exclusion of the "infinite"?

In the world of physics, things can get very very large, but not infinite. For example, if a physical model of some phenomenon predicts an infinite result in some circumstance, it signals a hard limit ...
user avatar
7 votes

How should one interpret modern mathematics if one doesn't believe in infinity?

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 ...
user avatar
  • 893
7 votes
Accepted

Is the axiom of infinity truly an axiom?

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 ...
user avatar
  • 20.6k
7 votes

Does science require the exclusion of the "infinite"?

First, let's concede there are two conceptions of the infinite. One is the potential and the other is the actual. As for excluding the infinite, I think it's fair to say that the answer is a ...
user avatar
  • 11k
6 votes

Are ordinal or cardinal infinities theories for real?

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 ...
user avatar
  • 3,407
6 votes

How does actual infinity (of numbers or space) work?

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 ...
user avatar
  • 40.8k
6 votes
Accepted

Are infinitesimals in the Newton and Leibniz calculus potential or actual?

Actual infinities collected into sets were not officially contemplated by (philosophizing) mathematicians until Cantor (with some anticipation by Bolzano) countered Aristotelian and scholastic ...
user avatar
  • 40.8k
5 votes

Does the concept of infinity mean we have already 'been'?

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, ...
user avatar
  • 2,871
5 votes

If there are infinite realities, do realities exist in which there are no infinite realities?

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 ...
user avatar
  • 204
5 votes

Limitless Space

In physics, infinity is usually a sign that our modelling of the physical situation has broken down; generally, the only kind of infinity that is allowed is the potentially infinite; this means that ...
user avatar
5 votes

Can infinity be defined?

Mathematics has a long and colourful history of dealing with infinity. If you know some real analysis, then you know that Cauchy (b. 1789) was the first to make rigorous our account of sequences and ...
user avatar
  • 1,135

Only top scored, non community-wiki answers of a minimum length are eligible