The Concept of the Infinite

[shunyadragon]

Well, then you've been hating on him too much to know that I'm not.



And how does this contradict the point that the past-eternal, physical existence of the universe is a metaphysical impossibility?


See the bold stuff? That's referring to metaphysical possibility. That has absolutely nothing to do with whether the universe is physical or not,



Well, thank goodness I don't just take your word for it. Why is it bogus? Sounds good to me!



That was perspicuous, lol. The response is completely misconceived. Great, you've emptied it out. In this possible world, Craig's metaphysical absurdities involving contradictions in transfinite arithmetic wouldn't arise with regard to people leaving and accommodating a potentially infinite amount of new guests when an actual infinite amount of guests are already in the hotel. In the new scenario, the hotel would never fill up! That's Craig's second philosophical argument: the metaphysical impossibility of forming an actual infinite via successive addition. Thanks for illustrating Hilbert's Hotel in a way that doesn't threaten Craig's argument at all!

I do not propose emptying anything. I propose beginning with a potentially infinite hotel that is empty. Now try and fill it. Hilbert's Hotel fits in my hotel.

Craig proposes that our (universe) physical existence cannot be 'past infinite.' That is not metaphysical issue.

[Tassman]

And how does this contradict the point that the past-eternal, physical existence of the universe is a metaphysical impossibility?

Given you acknowledge that actual infinities are mathematically possible metaphysical possibility is no longer the issue. A metaphysician can be mistaken in his deductions, just as a scientist can. But even if these are impeccable, he will not necessarily succeed if he argues correctly from premises that are unacceptable because they lack the necessary foundation in fact.

[mattbballman31]

I do not propose emptying anything. I propose beginning with a potentially infinite hotel that is empty. Now try and fill it. Hilbert's Hotel fits in my hotel.

Craig would have no problem saying you can't fill a hotel with a potentially infinite amount of rooms. His philosophical arguements don't even apply to a hotel with a potentially infintine amount of rooms in a hotel. What are you talking about?

Craig proposes that our (universe) physical existence cannot be 'past infinite.' That is not metaphysical issue.

Yes it is. Seriously. How is that not a metaphysical issue? The claims are,

1. An infinite temporal regress of events is an actual infinite
2. An actual infinite cannot exist.
3. Therefore, an infinite temporal regress of events cannot exist.

1 and 2 are metaphysical propositions, and so is the conclusion.

[mattbballman31]

Given you acknowledge that actual infinities are mathematically possible metaphysical possibility is no longer the issue. A metaphysician can be mistaken in his deductions, just as a scientist can. But even if these are impeccable, he will not necessarily succeed if he argues correctly from premises that are unacceptable because they lack the necessary foundation in fact.

I'm so lost right now. When Craig is saying that an actual infinite is mathematically possible, he saying that you shouldn't deny the "mathematical legitimacy to the actual infinite", like the intuitionists do. "Cantor's system and axiomatized set theory may be taken to be simply a universe of discourse, a mathematical system based on certain adopted axioms and conventions, which carries no ontological commitments." Ontological commitments is the modal realm of metaphysical possibility, or broad logical possibility, which is in terms of actualizability. Like David Hilbert said, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought . . . The role that remains for the infinite to play is solely that of an idea." So, just because you think there's mathematical legitimacy to the idea of actual infinites in infinite set theory, it does not follow that it is metaphysically possible to put into one-to-one correspondence each member of such a set and a concrete or a Platonically conceived abstract object. Craig doesn't even believe mathematical objects exist. Craig says, " . . . it is open to the mutakallim to hold that while the actual infinite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world."

[shunyadragon]

I'm so lost right now. When Craig is saying that an actual infinite is mathematically possible, he saying that you shouldn't deny the "mathematical legitimacy to the actual infinite", like the intuitionists do. "Cantor's system and axiomatized set theory may be taken to be simply a universe of discourse, a mathematical system based on certain adopted axioms and conventions, which carries no ontological commitments." Ontological commitments is the modal realm of metaphysical possibility, or broad logical possibility, which is in terms of actualizability. Like David Hilbert said, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought . . . The role that remains for the infinite to play is solely that of an idea." So, just because you think there's mathematical legitimacy to the idea of actual infinites in infinite set theory, it does not follow that it is metaphysically possible to put into one-to-one correspondence each member of such a set and a concrete or a Platonically conceived abstract object. Craig doesn't even believe mathematical objects exist. Craig says, " . . . it is open to the mutakallim to hold that while the actual infinite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world."

As Tassman noted, William Craig and David Hilbert may be mistaken in their deductions. Actual infinities are actually used in science to describe the physical nature of our existence. Actual infinities would descriptive of attributes of our physical existence within a greater potential infinity.

[shunyadragon]

Craig would have no problem saying you can't fill a hotel with a potentially infinite amount of rooms. His philosophical arguments don't even apply to a hotel with a potentially infintine amount of rooms in a hotel. What are you talking about?

Potential infinities would only exist as infinite sets within a greater potential infinity.

Yes it is. Seriously. How is that not a metaphysical issue? The claims are,

1. An infinite temporal regress of events is an actual infinite
2. An actual infinite cannot exist.
3. Therefore, an infinite temporal regress of events cannot exist.

1 and 2 are metaphysical propositions, and so is the conclusion.

No, they describe limits on the potential infinite nature of our physical existence as Craig proposes. Actually the claim that an actual infinity cannot exist is false.

[mattbballman31]

Potential infinities would only exist as infinite sets within a greater potential infinity.

Uh, potential infinities are finite sets.

No, they describe limits on the potential infinite nature of our physical existence as Craig proposes. Actually the claim that an actual infinity cannot exist is false.

Uh, yes. I don't care what they describe. I care about the nature of their descriptions. And your last point is completely irrelevant to that issue. Great! You think they can't exist. I disagree. But to say they can't exist is a metaphysical proposition about the impossibility of an actual infinite. The fact that you can't understand these simple distinctions means you just haven't read or understood the literature on the topic.

[shunyadragon]

Uh, potential infinities are finite sets.

Actual infinities are defined as 'completed' closed sets of infinities.

Source: https://www.iep.utm.edu/infinite/#H1

For purposes of doing mathematics and science, the actual infinite has turned out to be the most useful of the three concepts. Using the idea proposed by Bolzano that was mentioned above,

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Source: https://math.vanderbilt.edu/schectex/courses/thereals/potential.html

Completed infinity, or actual infinity, is an infinity that one actually reaches; the process is already done. For instance, let's put braces around that sequence mentioned earlier:
{ 1, 2, 3, 4, ... }
With this notation, we are indicating the set of all positive integers. This is just one object, a set. But that set has infinitely many members. By that I don't mean that it has a large finite number of members and it keeps getting more members. Rather, I mean that it already has infinitely many members.

We can also indicate the completed infinity geometrically. For instance, the diagram at right shows a one-to-one correspondence between points on an infinitely long line and points on a semicircle. (see reference for diagram) There are no points for plus or minus infinity on the line, but it is natural to attach those "numbers" to the endpoints of the semicircle.

Isn't that "cheating," to simply add numbers in this fashion? Not really; it just depends on what we want to use those numbers for. For instance, f(x)=1/(1+x2) is a continuous function defined for all real numbers x, and it also tends to a limit of 0 when x "goes to" plus or minus infinity (in the sense of potential infinity, described earlier). Consequently, if we add those two "numbers" to the real line, to get the so-called "extended real line," and we equip that set with the same topology as that of the closed semicircle (i.e., the semicircle including the endpoints), then the function f is continuous everywhere on the extended real line. This has some advantages in advanced mathematics: The topology of the closed semicircle is compact and metrizable. Compact metric spaces have very nice topological properties; for instance, every sequence has a convergent subsequence. Even if we're really just interested in the properties of the ordinary (finite) real numbers, we can discover and prove some of those properties more easily by viewing that set of numbers as a subset of this larger, compact metric space.

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Uh, yes. I don't care what they describe. I care about the nature of their descriptions. And your last point is completely irrelevant to that issue. Great! You think they can't exist. I disagree. But to say they can't exist is a metaphysical proposition about the impossibility of an actual infinite. The fact that you can't understand these simple distinctions means you just haven't read or understood the literature on the topic.

Read my posts again and respond intelligently. Math is descriptive of our physical nature and math is not material objects that exist nor not exist. Actual infinities exists as descriptions of the physical properties of our physical existence.

If your care what they describe, they actually do exist as descriptions of attributes of our physical universe.

Source: https://plus.maths.org/content/do-infinities-exist-nature-0

Do infinities exist in nature?
By
Marianne Freiberger and Rachel Thomas
Submitted by Marianne on September 26, 2013
What would you see if you came to the edge of the Universe? It's hard to imagine so it's tempting to conclude that the Universe doesn't have an edge and therefore that it must be infinite. That's not a necessary conclusion however. There are things that are finite in extent but still don't have an edge, the prime example being the surface of a sphere. It's got a finite area but when you walk around on it you'll never fall over an edge. The question of whether the Universe is finite or infinite is one that still hasn't been answered, and there are mathematical models that allow for both possibilities. More generally, the question of whether any infinite quantities can arise in the Universe is a deep one. In April this year philosophers, cosmologists and physicists came together at the University of Cambridge, as part of a conference series on the philosophy of cosmology, in order to discuss it. Plus went along to find out more (and you can also listen to the interviews we did in our podcast).

Infinity that doesn't bite
John D. Barrow

People have been studying infinity and its relation to reality for a long time. "The idea of studying infinities in physics really began with Aristotle," says the Cambridge cosmologist John D. Barrow. "Aristotle made a clear distinction between two types of infinity. One he called potential infinities and he was quite happy to allow for those to appear in descriptions of the world. These are just like lists that never end. The ordinary numbers are an example; one, two, three, four, five, and so on, the list goes on forever. It's infinite, but you never reach or experience the infinity. In a subject like cosmology, there are lots of infinities like that and most people are quite happy with them. For example, the Universe might have infinite size; it might have an infinite past age, it might be destined to have an infinite future age. These are all potential infinities, so they don't bite you as it were, they're just ways of saying that things are limitless, they're unbounded, like that list of numbers."

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[mattbballman31]

Actual infinities are defined as closed sets of infinities.

You really are dense.

YOU SAID THE POTENTIAL INFINITE WAS AN INFINITE SET, YOU MORON.

Then I said that the potential infinite was a finite set.

And your response is to mention that ACTUAL INFINITIES ARE DEFINED AS CLOSED SETS OF INFINITIES????

NO DUH! It's like I'm talking to a darn wall. I wasn't talking about actual infinites, moron. I was responding to your claim that potential infinites are infinite sets. They aren't. Got that, moron? So, your dumb IEP quote is idiotically irrelevant, moron. Learn to read and stop wasting my time.

Read my posts again and respond intelligently.

Why don't you follow your own genius advice?

Math is descriptive of our physical nature and math is not material objects that exist nor not exist. Actual infinities exists as descriptions of the physical properties of our physical existence.

Did you even read my last post, moron? I said that . . . Craig . . . does not deny the mathematical legitimacy of the actual infinite. THAT MEANS I DON'T DENY THAT MATH IS DESCRIPTIVE OF PHYSICAL NATURE, MORON. How in the freaking world does this have anything to do with the idea that " . . . it is open to the mutakallim to hold that while the actual infinite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world."

And then you just quote these huge block quotes I've already addressed, dunce. Marianne Freiberger and Rachel Thomas don't say ANYTHING addressing the point. And Barrow is only talking about POTENTIAL infinities. You either really suck at reading and comprehension or you're so blinded by your own bias that any semblance of objectivity has evaporated into the stratosphere and beyond.

I can't wait what nonsense you'll spew next! Stay tuned!

[shunyadragon]

You really are dense.

YOU SAID THE POTENTIAL INFINITE WAS AN INFINITE SET, YOU MORON.

Then I said that the potential infinite was a finite set.

And your response is to mention that ACTUAL INFINITIES ARE DEFINED AS CLOSED SETS OF INFINITIES????

NO DUH! It's like I'm talking to a darn wall. I wasn't talking about actual infinites, moron. I was responding to your claim that potential infinites are infinite sets. They aren't. Got that, moron? So, your dumb IEP quote is idiotically irrelevant, moron. Learn to read and stop wasting my time.



Why don't you follow your own genius advice?



Did you even read my last post, moron? I said that . . . Craig . . . does not deny the mathematical legitimacy of the actual infinite. THAT MEANS I DON'T DENY THAT MATH IS DESCRIPTIVE OF PHYSICAL NATURE, MORON. How in the freaking world does this have anything to do with the idea that " . . . it is open to the mutakallim to hold that while the actual infinite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world."

And then you just quote these huge block quotes I've already addressed, dunce. Marianne Freiberger and Rachel Thomas don't say ANYTHING addressing the point. And Barrow is only talking about POTENTIAL infinities. You either really suck at reading and comprehension or you're so blinded by your own bias that any semblance of objectivity has evaporated into the stratosphere and beyond.

I can't wait what nonsense you'll spew next! Stay tuned!

I apologize for the typo. It is actual infinities that are 'complete' closed sets. They are description of attributes of our physical existence.

I correctly described it in the next post. In this context actual infinities do indeed exist in nature.

[mattbballman31]

I apologize for the typo. It is actual infinities that are 'complete' closed sets. They are description of attributes of our physical existence.

I correctly described it in the next post. In this context actual infinities do indeed exist in nature.

Wow. Admission of error. We're making progress! There's hope for you yet.

The quote you gave doesn't show that actual infinities exist in nature at all. It's like you copy/past some mathematical explanation and are unable to connect the dots to metaphysical reality. All the quote does is define, mathematically, an actual infinite (something Craig affirms), extend such a definition geometrically from points on a line to points on a semicircle (something Craig wouldn't deny), and show how it wouldn't be 'cheating' if such a topological extension is done in geometry. Wow. Cool. Oooh! Ahhh! Shuny can quote blocks of irrelevant bilge. Craig denies none of this, and it has nothing (absolutely nothing!) to do with saying that actual infinities exist in nature.

Oh, and because you're such a Craig-expert, quote me exactly where Craig addresses the very point your clumsily trying to target that because one can successfully utilize actual infinites relative to topological geometrical structures for whatever the mathematician 'wants to use the numbers for', that this therefore implies the begged metaphysical question of whether the topological structure of space itself is metaphysically identical to the structure as delineated in geometry. He addresses it! It's in his literature on the topic! And rather than hold your hand and quote it for you, either admit that you don't have a sufficient grasp on the argument as Craig has presented it since freaking 1979 or that you're an embarrassing armchair block-quote copy and paster whose only concern is the appearance of sophistication, an appearance that itself is a laughable disguise. I'm so sick of pseudo-intellectuals like you thinking that you can just throw some stupid gotchya-spaghetti against the wall, and you actually are convinced that you've GOT CRAIG WITH HIS BACK AGAINST THE WALL. It's absurd. He's explored the darn argument almost into the ground, and these silly 'arguments' you bring up would just be laughed out of your normal philosophy conference.

[shunyadragon]

This article by Morriston addresses the problems with Craig's arguments concerning actual infinities in great detail.

Source: http://spot.colorado.edu/~morristo/craig-on-the-actual-infinite.pdf

I do not believe that Craig has succeeded in
proving that actually infinite sets are impossible. As far as this particular line of
argument is concerned, I shall try to show that it remains an open question
whether the past could consist in a beginningless series of events.

© Copyright Original Source

My previous references confirm that actual infinities do exist in nature.

[JonathanL]

My previous references confirm that actual infinities do exist in nature.

No they didn't. I read through the whole article you referenced in post #53, and not once did I see the authors make the claim that actual infinities have been confirmed to exist in nature.

[shunyadragon]

No they didn't. I read through the whole article you referenced in post #53, and not once did I see the authors make the claim that actual infinities have been confirmed to exist in nature.

Yes it demonstrated Craig's assertions as false, as well as other references describing actual infinities in nature. More to follow.

Self imposed ignorance leers its ugly head.

Again this article discusses actual infinities as they exist in describing their properties using math to describe nature.

[JonathanL]

Yes it demonstrated Craig's assertions as false, as well as other references describing actual infinities in nature. More to follow.

Self imposed ignorance leers its ugly head.

Again this article discusses actual infinities as they exist in describing their properties using math to describe nature.

Can you please bold the part that you misunderstood to confirm the existence of actual infinities in nature?