The set introduced in the Axiom of Infinity is not necessarily the Natural numbers nor is it necessarily an isomorphism of the Natural numbers.
The Axiom of Infinity says that there is an inductive set (one that contains the empty set and is closed under the successor operation). The set of Natural Numbers is then *defined* to be the smallest inductive set (or, alternatively, the intersection of all inductive sets).
From just the Axioms of Pairing and of Union, it is possible to define what it means to be a natural number (a non-limit ordinal that contains no limit ordinal) and thereby to talk about the *class* of natural numbers. The import of the Axiom of Infinity is that this class is actually a set and not a proper class.
It is insufficent for that purpose. In order to define a morphism to the Natural numbers, you have to define the Natural numbers (or something that you would consider to be isomorphic to the Natural numbers that you then call the "natural numbers"). Otherwise, it is an morphism to... ??? In this case, the Natural numbers requires an initial element (conveniently that would be the empty set) and a successor function x U {x}. Now you can try to define your morphism.... maybe. I leave that to you to work out.
Once again, it is perfectly possible to talk about the collection of natural numbers as a *class* prior to the Axiom of Infinity. Also, the morphisms are also perfectly possible to discuss and formulate without it. ALL that the AI does is say that there is a *set* rather than a proper class of natural numbers.
Yes, and when you trace that chain of causality you arrive at the first mover (you).
Not necessarily. For example, suppose a new guest arrives every day. They are told (by the current manager) to go to the first room, and change to the next higher room every night. The managers change every 1000 days and are told about the instructions for new guests.
This has been going on literally forever. So there are infinitely many guests (one for each room), each was told what to do by the manager at the time they arrived. But each guest and each manager is given instructions.
The chain of causality for each guest goes back to a manager, and then back through infinity many managers.
"The guest are moved" by you.
Not in the new scenario. They are directed to move each later night by one manager, but no single manager told all the guests how to move.
If you have an infinite regress of explanation, then you haven't actually explained why any of the guests move.
They have literally always been moving. No further explanation is required.
Language can be tricky. I'll leave you to consider the implications of your language carefully. This notion of binding causality to time and binding time to the universe is predominant in physical science as opposed to purely metaphysical considerations.
Indeed, language can be tricky, especially in these matters. There are hidden assumptions about time in much of our language that may not be appropriate.
Can you give a single 'metaphysical cause' that is not a physical cause?
If there were no "beginning" to the Earth (when considered as a static 3-dimensional object), you would be unable to define a 2-dimensional manifold describing its surface. You would say, "There is no surface."
So you don't understand the analogy.
o use speculative models to oust main theory. If you don't understand the objection, I don't think I can make it any more clear. Sorry.
Speculative models with a LOT of observational support.
Your argument is invalid.
Given:
1. if p, then q
2. ~p
_________________
It does not follow that:
3. ~q
Moreover, if you are trying to say something substantial about the truth of the premises, then don't claim you aren't. And if you aren't, then don't claim you have!
That is NOT my argument. My argument is
1. If p, then q
2. We don't know whether p is true or not.
BUT
the claim is made (in the OP) that we can conclude q.
My claim is that this argument is unsound because we don't know whether p is true or not. If it is NOT true, the the conclusion of q is invalid.
Putting aside your... characterization of the OP. Arguments need not be "convincing" in order to be valid or sound. You are under no compulsion to believe or disbelieve the truth of the premises, the validity of the argument, or the soundness of the conclusion.
The soundness of the argument requires that the assumptions be true. if they are not known to be true, the argument is unsound.
I don't think the question of whether or not the universe has a beginning (from a philosophical point of view) was answered.
I also don't think people are going to agree about it. They will believe what they want to believe.
Good! that is precisely my point! The conclusion that the universe has a beginning cannot be established by philosophy alone. In the same way, whether there is a 'first cause' cannot be determined by philosophy alone.