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39 votes
Accepted

Why would infinite monkeys not produce the works of Shakespeare?

Yes, the monkeys will do it. No, they don't have to. It's mathematically true that after removing all logistical constraints - which is what we mean when we say there are infinitely many monkeys, ...
Zayn's user avatar
  • 640
33 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 ...
Noah Schweber's user avatar
21 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 ...
Conifold's user avatar
  • 43.5k
17 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 ...
Conifold's user avatar
  • 43.5k
13 votes
Accepted

Can a totally ordered set with a last element but no first element exist, or is this contradictory?

Can a totally ordered set with a last element but no first element exist, or is this contradictory? Taking the usual mathematical definition of total order, and taking "last element" to ...
John Bollinger's 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 ...
Conifold's user avatar
  • 43.5k
11 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, ...
Adam Sharpe's user avatar
  • 3,864
11 votes

Is infinity a concept or a word empty of meaning?

Your comment seems to be the nub of your problem: I cannot think about infinity because my finite meaning cannot grasp even conceptually non finite objects, despite that there are perhaps infinite ...
Rushi's user avatar
  • 3,393
10 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 ...
Jo Wehler's user avatar
  • 33.6k
10 votes
Accepted

Is infinity a concept or a word empty of meaning?

Mathematics shows that we can make a one to one correspondence [of the] natural numbers with [the] even numbers. This is not right If there is a last number. There are infinite sets that look exactly ...
ac15's user avatar
  • 1,793
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 ...
kutschkem's user avatar
  • 2,620
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 ...
niels nielsen's user avatar
7 votes
Accepted

Is there such a thing as ฯ‰th-order infinitary logic?

Infinitary logics like L(๐œ”1,๐œ”) (the simplest and best-behaved of the bunch) and higher-order logics like second-order logic extend first-order logic in different directions. Infinitary logics extend ...
Noah Schweber's user avatar
7 votes

Is the B-theory of time only compatible with an infinitely renewing cyclical reality?

No, an infinite sequence of events need not repeat, just as an infinite series of numbers need not repeat (e.g. the digits of pi). Your infinitely long rope example is obviously flawed, because we ...
Eric Smith's user avatar
6 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 ...
J D's user avatar
  • 28.5k
6 votes

Can a totally ordered set with a last element but no first element exist, or is this contradictory?

Take the negative integers. But remember that arbitrarily large numbers donโ€™t require an infinite item. There are arbitrarily large and small integers, but no infinite ones.
gnasher729's user avatar
  • 5,647
6 votes

How to understand the notion of majority when comparing infinite sets?

This is already a problem even for simple probability distributions in the real numbers, such as the normal (Gaussian) distribution. The reals are infinitely dense, so we cannot assign individual ...
Kevin's user avatar
  • 2,148
6 votes

Is infinity a concept or a word empty of meaning?

Literally, infinity is both a concept and a word- that should be clear to you from the fact that you have been thinking about the concept and have typed the word more than once in your post. Infinity ...
Marco Ocram's user avatar
  • 23.6k
5 votes

What is the difference between an Ordinal number and a Cardinal number?

Actually, the difference cal already be seen for finite numbers, although they get really manifest only in the infinite numbers. Cardinals are about the question "how many". For example, there are ...
celtschk's user avatar
  • 1,559
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 ...
Mozibur Ullah's user avatar
5 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 ...
Conifold's user avatar
  • 43.5k
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 ...
Sputnik's user avatar
  • 1,155
5 votes

Does science require the exclusion of the "infinite"?

No, there's no need whatsoever to exclude the infinite from science. The gold standard for a scientific hypothesis is that the hypothesis is consistent with all known observations, successfully ...
Tanner Swett's user avatar
5 votes

Why would infinite monkeys not produce the works of Shakespeare?

The difference is between in principle and in practice. If you did have an infinite number of monkeys typing at random for eternity they would produce the works of Shakespeare. However, even if every ...
Marco Ocram's user avatar
  • 23.6k
5 votes

What is the state-of-the-art of formal definitions of God?

It is well-known that neither a biggest cardinal number nor a biggest ordinal number exist. I assume that also Cantor, who invented transfinite set theory, was aware of this result from his theory. I ...
Jo Wehler's user avatar
  • 33.6k
5 votes
Accepted

What is the state-of-the-art of formal definitions of God?

Horsten[16] (see also Barton[22]) compares the use of the concept of absolute infinity in theology to its use in theories of proper classes. (Horsten also happens to have written the current SEP ...
Kristian Berry's user avatar
4 votes

How many numbers does it take to describe conscious reality?

Does the uncountability of the "real" numbers imply that consciousness is much larger than the brain? No. The 2 things are wholly unrelated. You are essentially requiring that the physical brain ...
Alex's user avatar
  • 1,828
4 votes

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

Axiom A2: If set A is a proper subset of set B, then A has a smaller cardinality than set B is intuitive. Correct; since ancient time [see Euclid's Elements] it was assumed : Common notion 5. The ...
Mauro ALLEGRANZA's user avatar

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