Actual Infinity in Reality?

[SteveII]

Since there are new people joining the discussion, here is a summary of problems as I see it. 

INFINITE SET THEORY
Some have suggested that the area of infinite set theory in mathematics is an indication that an actual infinite is possible. So this is how that argument goes.

1. An actual infinite consists of real (not abstract) objects.
2. The Axiom of Infinity assumes an abstract object that is an actual infinite set
3. Paradoxes, contradictions, and absurdities (such as below) can be dismissed because we can apply infinite set theory parameters/restrictions to the actual objects
     Hilbert's Hotel
     Galileo's paradox
     Ross–Littlewood paradox
     Thomson's Lamp paradox
     Zeno's Paradoxes
4. Therefore actual infinities are possible.

Since (2) is of the class of axioms that are not self-evident, they are assumptions on which further mathematical equations can be developed (useful in calculus for example). To be clear, this axiom is not reasoned into--it is just assumed as a foundation for the subset of infinite set theory in mathematics. See this earlier post. 

This is CLEARLY a question-begging argument and therefore invalid. 

B THEORY OF TIME
Another argument that has been made is that if the B Theory of Time is correct, spacetime is infinite in extent. But there is nothing in the theory that says our spacetime is infinite in the past. To get that, you must also posit an infinite cosmology model. But such models are not thought to be the best candidates for our universe, so, while possible (broadly speaking), there are not good reasons to believe this to be the case. But, such a combination of theories seems possible, so then doesn't that show that an actual infinity is possible. No, not at all.

Under any theory of time there is some sequence that is countable whether you call it causes/connection/light cones/changes in entropy/states of affairs/or whatever. I'll call it causal connections (but insert whatever you want).  Any timeline would show that the causal connections that created the present were preceded by causal connections which were preceded by causal connections for an infinite series in the prior-to direction. If you posit an infinite number of these causal connection going back, you have a problem. How could we have traversed through an infinite number of sequential causal connections to get to the one that caused the present (causal connection 5, 4, 3, 2, 1, 0)? There will always have to be infinite more causal connections that still need to happen. We will never arrive at the present.

To illustrate it with a thought experiment, imagine a being who is counting down from eternity past to the present: 5, 4, 3, 2, 1, now. How is that possible? Wouldn't he have an infinite amount more numbers to get through to get down to 3, 2, 1? If you insist that this could be done, why didn't he get done 1000 years earlier or for that matter, an infinite time ago? 


AGAINST AN ACTUAL INFINITE
A positive argument against an actual infinite is:

5. An actual infinite consists of real (not abstract) objects.
6. In 100% of our experiences and 100% of our scientific inquiries, quantities of real objects can have all the operations of addition, subtraction, multiplication and division applied to them. 
7. As the following paradoxes show, these operations cannot be applied to the concept of an actual infinite without creating contradictions and absurdities
     Hilbert's Hotel
     Galileo's paradox
     Ross–Littlewood paradox
     Thomson's Lamp paradox
     Zeno's Paradoxes
8. Therefore an actual infinite of real objects is metaphysically impossible. 

Infinite set theory is not a defeater for (6) because infinite set theory is not itself a conclusion derived from a logical process. To defeat (6) you have to give logical reasons why we should expect an infinite quantity of objects to behave fundamentally different than a finite quantity of objects.