That is not the mathematical definition of a set. The mathematical definition of a set is that it obeys the axioms of the set theory in which we are working. The most commonly-used set of axioms is Zermelo-Frankel - ZF. The concept of 'collection' does not form part of those axioms.
But even if we were to try to use the definition you suggest, it would be incorrect to say that infinite sets are not well-defined. In mathematics the words 'well-defined' have a very specific meaning, and they only apply to functions, not properties (aka relations). We say that a function is 'well-defined' if, using the definition to apply it to an element of its domain, there is a unique object that is the image of that element under that function. The notion of being a set, or of having finite cardinality, is a property, not a function, so the notion of 'well-defined' is not relevant. — andrewk
If you really dislike the concept of infinity, all you need do is reject the 'Axiom of Infinity', which asserts the existence of a set that can be thought of as the set of natural numbers. Without such an axiom, we can have natural numbers as large as we wish, but there is no such thing as the set of all natural numbers. Such an approach to mathematics is consistent, and some people try to limit themselves to that. The trouble is that it is that axiom that gives us the tool of Proof by Mathematical Induction. Without it, there is an enormous volume of important results that we would not be able to us. — andrewk
As you point out, we lose quite a lot of mathematics by dropping the Axiom of Infinity. And to me the foregoing arguments for doing so are ridiculous since the claim is contradictions occur with the axiom (they don't). Even Ultrafinitists don't claim that provable contradictions appear, so the justification for dropping the Axiom just looks like philosophical bias more than anything. — MindForged
This works for a logical theory in which the only objects in the domain of discourse are natural numbers. In that case, we can just use the following axiom of induction:All that mathematical induction requires is that what is true of one number is true of the next, and therefore true of all the following numbers — MU
— tim wood
There cannot have not been infinitely many paths TAKEN, there are only infinitely many possible paths that could potentially be taken, but it is impossible to actually follow them - no matter how long we have to try. So these paths exist in the abstract, but not in the real world. — Relativist
Arguably: no, they aren't real.Why do they have to be taken to be real? If they're not taken are they not real? .... Each is a possible path. It's the "taken" you object to? But whenever was a clock attached to a number?
But even if we were to try to use the definition you suggest, it would be incorrect to say that infinite sets are not well-defined. In mathematics the words 'well-defined' have a very specific meaning, and they only apply to functions, not properties (aka relations) — andrewk
The trouble is that it is that axiom that gives us the tool of Proof by Mathematical Induction. — andrewk
finite universe is impossible, because it'd have borders with nothingness, which is the logical definition of crazy — SnoringKitten
Now, have Achilles stop to catalogue the infinite subdivisions - forget Planck - that would be attaining to actual infinity. I believe in an actual infinity btw! — SnoringKitten
You have defined a new term in relation to sets - 'fully defined'. What then?Ok let’s use the language ‘fully defined’. A set is only fully defined once we have listed all its members. Clearly infinite sets are not fully defined yet maths tries to treat them in the same way as a finite set (which is fully defined). — Devans99
I think the mathematicians have the definition of Point wrong:
“That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, volume or any other dimensional attribute’
https://en.m.wikipedia.org/wiki/Point_(geometry)
If a point has no length it does not exist so the definition is contradictory.
A point must has length > 0 else it does not exist. With this revised definition of a point we can see that the number of points on any line segment is always a finite number rather than Actual Infinity. — Devans99
However this doesn't work if we want to have objects in our domain of discourse other than natural numbers, because then we need to add a condition 'x is a natural number' to the above induction axiom, which requires referring to the set of natural numbers, whose existence cannot be asserted without the axiom of infinity, or some equivalent.. — andrewk
I think this issue of the axiom of infinity may be related to that of omega-completeness, which is about whether there may be natural numbers other than those we get by adding 1 to 0 a finite number of times, ie 'non-standard' natural numbers. Omega-completeness is a very interesting subject, but it usually gets my head all muddled when I try to think about it, if I haven't done so in recent times. — andrewk
The premise that we start with is that Before The Beginning there was Absolute Nothing. There was no Energy, and there was no Matter, and there was no Space. — SteveKlinko
The premise that we start with is that Before The Beginning there was Absolute Nothing. There was no Energy, and there was no Matter, and there was no Space. — SteveKlinko
I don’t think you can start with that premise:
- Something can’t come from nothing
- So something must have always existed
- So the state of ‘Nothingness’ is impossible
- If something is permanent it must be timeless (proof: assume base reality existed eternally - the total number of particle collisions would be infinite - reductio ad absurdum)
- So base reality must be timeless (to avoid the infinities)
- Time was was created inside this base reality — Devans99
Something can't come from nothing" is an unproven Belief when it comes to the beginning of everything. — SteveKlinko
Something can't come from nothing" is an unproven Belief when it comes to the beginning of everything. — SteveKlinko
I’m basing my argument on common sense and naturalism - not referencing any particular rule of physics.
- if you define nothing as no matter, energy, space or dimensions
- then it’s pretty clear ‘can’t get something from nothing’ holds
- so it follows something has existed always — Devans99
What is this Naturalism? How do you know Naturalism holds before the beginning. — SteveKlinko
- if you define nothing as no matter, energy, space or dimensions
- then it’s pretty clear ‘can’t get something from nothing’ holds
- so it follows something has existed always — Devans99
I assert that ‘something from nothing’ is a magical proposition so we can exclude from our investigations of the origin of things. — Devans99
That's self-contradictory. A beginning has no predecessor, or it's not the beginning.But when it comes to before the beginning nobody knows anything. — SteveKlinko
Now imagine a Square that is the smallest Square that is not equal to Zero. This thought sends your mind into an endless recursive loop of the Square getting smaller and smaller and we soon realize that it is impossible to imagine such a smallest Square. One thing we can say is that this Square is Infinitely small but is still a Square. In general mathematics this would be called a differential Square or an infinitesimal Square. — SteveKlinko
What is this Naturalism? How do you know Naturalism holds before the beginning. — SteveKlinko
Naturalism is the exclusion of magic from our consideration of the physical sciences.
I assert that ‘something from nothing’ is a magical proposition so we can exclude from our investigations of the origin of things. — Devans99
But how can you know that Naturalism holds before the Beginning? — SteveKlinko
I am not inclined to drop the idea that the natural numbers are infinite, only the idea that the infinite natural numbers are a set. — Metaphysician Undercover
As andrewk indicates, if your axiom states that a set may be finite or infinite, then that is what is the case in that axiomatic system. The problem that I see, is that the way "set" is used by mathematicians, as a closed, bounded object, the possibility of an infinite set is precluded. Sets are manipulated by mathematicians, as bounded objects, but an infinite set is not bounded like an object, and therefore cannot be manipulated like an object. This calls into question the understanding of "infinite" which is demonstrated by this axiom of infinity, which stipulates that the infinite natural numbers are a "set".
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