Actual Infinity in Reality?

[GrandizerII]

[SteveII]

[polymath257]

[polymath257]

I'd point out that *finite* sets in mathematics are just as 'abstract'. mathematics is the study of formal axiom systems. So of course it is all abstract.

The idea that axioms are 'intuitively obvious' is another outdated idea. Euclid tried that with Euclidean geometry, but we have found that his system, even of geometry, was far from unique. Non-euclidean geometry is equally consistent, but gives different answers.

Which leads to the point: when there are different axioms systems that are all consistent, there is nothing to say which is correct and which is not correct except to go to observation and testing.

So, the fact that introducing actual infinities does bring a contradiction shows that there is no *logical* reason to exclude actual infinities.

And I agree--the actual existence of actual infinites has not been proven. But that isn't my claim. My claim is that there is no *logical* issue with them and that they should be considered as one *possibility*. And that is quite enough to destroy the Kalam argument. There is no *contradiction* with having an infinite regress of causes. It is internally consistent and so cannot be dismissed out of hand.

[GrandizerII]

Nonlogical axioms? Way to diss mathematics, Steve. You will say anything for God to make sense. Interestingly enough, Cantor was a believer and was strengthened in his faith because of actual infinities.

Edit: Just read the quotes in full. Nonlogical meaning different from what I initially thought. Still, no logical contradictions

[SteveII]

THEREFORE, because infinite sets are a results of non-logical axioms (assumptions) that are not self-evidently true AND, the term 'exist' is not the same in mathematics, I stand by my claim that infinite sets in mathematics are not an indication that an actual infinite series or objects can exists in reality.

But the math shows that such assumptions lead to no internal contradictions: they are logical possibilities. And that is the whole point: that there is no *logical* obstacle to these being real.

The math is based on axioms (assumptions). It is question begging (circular reasoning) to say it is proof that they are logical possibilities. You have assumed an actual infinite in order to do further math with it. So, it gives no help to the argument that an actual infinity can exists. Therefore need we turn to something other than math:

1. You cannot get to infinity by successive addition.
2. You get absurdities when you propose an infinite number of actual objects (Hilbert's Hotel).
3. You get contradictions about how many squares and square roots there must be (Galileo's paradox)
4. Is the vase full or empty in the Ross–Littlewood paradox?
5. Is the lamp on or off in the Thomson's lamp paradox?
6. It seems we cannot traverse even a finite distance in Zeno's paradoxes

These coupled with the fact that we don't have anything in the real world that could be an actual infinite leads a rational person to the believe that an actual infinite of real objects is not possible.

[polymath257]

But the math shows that such assumptions lead to no internal contradictions: they are logical possibilities. And that is the whole point: that there is no *logical* obstacle to these being real.

The math is based on axioms (assumptions). It is question begging (circular reasoning) to say it is proof that they are logical possibilities. You have assumed an actual infinite in order to do further math with it. So, it gives no help to the argument that an actual infinity can exists. Therefore need we turn to something other than math:

1. You cannot get to infinity by successive addition.
2. You get absurdities when you propose an infinite number of actual objects (Hilbert's Hotel).
3. You get contradictions about how many squares and square roots there must be (Galileo's paradox)
4. Is the vase full or empty in the Ross–Littlewood paradox?
5. Is the lamp on or off in the Thomson's lamp paradox?
6. It seems we cannot traverse even a finite distance in Zeno's paradoxes

These coupled with the fact that we don't have anything in the real world that could be an actual infinite leads a rational person to the believe that an actual infinite of real objects is not possible.

On the contrary, the fact that we do not get inconsistencies from these axioms shows there is no logical problem with them.

We know of many axiom systems that *are* self-contradictory. The theory of infinite sets is not one of them.

1. Irrelevant. That isn't the mechanism for getting infinite sets.
2. Not absurdities, just counter to intuition derived from the study of finite things.
3. Not a contradiction. Again, just counter to intuitions derived from the study of finite things.
4. Not well defined. An impossible task due to relativity.
5. Not well defined. Task impossible to do because of relativity.
6. Resolved because both space and time are infinitely divisible: see algebra and calculus.

the lack of coherent arguments against actual infinities and the fact that they are not self-contradictory is enough to show they are *possibilities*.

Again, whether they are present in the real world is not known. But there is no *logical* issue. Paradoxes because people think in terms of finite sets, yes. But no contradictions.

---

Once again, Steve, you cannot get Graham's number (which is finite) via successive additions for any operations, as fast as they could be, in the current age of the universe.

Do you think Graham's number can exist as a logical possibility?

[GrandizerII]

I dont know about some of the other paradoxes as Im not too familiar with them, but regarding Zenos continuity paradoxes, supposing it is a problem for motion, this universe might be both finite and discrete anyway ... maybe. Still doesnt mean infinite divisibility is illogical. But Zeno also had paradoxes that seem to show that motion was impossible even with discrete objects. Does this mean motion is illogical, Steve?

Also Grahams number is a beast. Youll find it fascinating when you read up on it. Fun read.

[Jenny A]

Steve,

Sorry to be late in the conversation.  But I really am puzzled as to why you think ifinities pose a logical contradiction as opposed to being hard to show in reality. 

I think the hotel idea sounds intuitively wrong to you merely because it's mathematical.   Infinite odd and infinite even numbers may feel wrong but isn't.  Think of it without numbers.  Suppose there is an infinite number of smoking rooms and an infinite number of non smoking rooms.  Two infinities of different kinds of things.  Together, both are infinite, yet you have twice as many room choices should you  be ambivalent to smoke.  

If that's still too close, consider infinite cats and infinite dogs .  Both infinities together are infinite pets.  Still infinity but with more choices.  

No logical contradiction.  Just more kinds in the omega.

[GrandizerII]

Steve,

Sorry to be late in the conversation.  But I really am puzzled as to why you think ifinities pose a logical contradiction as opposed to being hard to show in reality. 

I think the hotel idea sounds intuitively wrong to you merely because it's mathematical.   Infinite odd and infinite even numbers may feel wrong but isn't.  Think of it without numbers.  Suppose there is an infinite number of smoking rooms and an infinite number of non smoking rooms.  Two infinities of different kinds of things.  Together, both are infinite, yet you have twice as many room choices should you  be ambivalent to smoke.  

If that's still too close, consider infinite cats and infinite dogs .  Both infinities together are infinite pets.  Still infinity but with more choices.  

No logical contradiction.  Just more kinds in the omega.

Yep both are without ends so naturally the total set will be without end as well.