37
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, ...
- 620
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 ...
- 3,211
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 ...
- 42.5k
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 ...
- 42.5k
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 ...
- 282
16
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 ...
- 42.5k
15
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 ...
- 24k
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 ...
- 648
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 ...
- 42.5k
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 (...
- 24k
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, ...
- 3,784
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 ...
- 2,232
8
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. ...
- 209
8
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 ...
- 24k
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 ...
- 7,772
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 ...
- 42.5k
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 ...
- 22.7k
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 don’t require an infinite item. There are arbitrarily large and small integers, but no infinite ones.
- 5,243
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 ...
- 1,733
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 ...
- 1,145
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 ...
- 813
4
votes
Finity to Infinity?
It would be correct to say: "An infinite set has proper subsets which are infinite again."
Otherwise trivial counter examples exist as Mauro points out.
The concept of infinity as detected by ...
- 24k
4
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 ...
- 1,511
4
votes
Could it be possible to refute Cantor's findings about multiple infinities on the basis of a radical new concept, that decimals are not numbers?
Set Theory proves that there are multiple infinities of sets. I don't see how showing that decimals (reals?) are not numbers has anything to do with it. Sets are the only objects in Set Theory; what ...
- 6,466
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 ...
- 1,808
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