RE: Actual Infinity in Reality?
February 26, 2018 at 6:57 pm
(This post was last modified: February 26, 2018 at 7:20 pm by polymath257.)
(February 26, 2018 at 6:43 pm)SteveII Wrote:(February 26, 2018 at 2:55 pm)polymath257 Wrote: 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?