Infinite Regress

[Clizby Wampuscat]

Division by 0 isn't defined. In order to get to a point where you can do calculus on the real or complex numbers you first have to build up the set theoretic and algebraic structures. But the required axioms don't allow for division by 0, because for ANY number a, if a|b iff there exists a number c s.t. a*c=b
Thus, if a=0, then c=0 and b=0 and the necessary structure doesn't hold. This structure is required for the necessary inverse properties.

Also keep in mind that it was widely believed for a long time that 0 wasn't a number, and that negative numbers were nonsense. What is or isn't a number, practically speaking, is a matter of definitions. More philosophically, but still mathematically, it has more to do with preferences and the philosophical prejudices one has.

Finally, I'm not sure what your point is when you include this: lim x-->infinity (1/x) = 0
After, all, as you know, lim x-->0 (1/x) = infinity. So it isn't just a limiting process but also the limit point itself, which is a bit of a problem (from a rigorous approach) if this point can't even be approached in the reals let alone be fined as the unique limit. These and similar issues are one thing that motivates the extension of R to to R* (the extended reals) and the associated algebraic properties of -infinity and infinity.

How does this relate to my comment?

 

[LegionOnomaMoi]

Ok, so how does this relate to my comment.

How does this relate to my comment?

I'll be more blunt then. In short, I was asking in a round about way why you would treat infinity like a number implicitly while claiming it isn't through an argument that seems to rely on completely irrelevant claims about number systems that are also not really true. Put even more concisely, and in another attempt to be helpful rather than critical, I was asking that you flesh out your position a bit more, either clarifying what you mean by "not a number" in a manner that doesn't require implicit appeals to structures defined on sets (because the same argument can be made for any "number" or most "numbers" in the case of Boolean fields and algebras).

And just to give you some context:

Spoiler: Context: May be skipped

One of the problems in discussions such as these, at least in my experience, is that many people will:
-make assertions about what is or isn't possible
-claim that X must be true, therefore Y can't be
-informally "show" that something follows via combinations of propositional-like logic, argument structures, informal reasoning methods, appeals to common sense, invalid inferences, explict "assumptions" stated as clearly necessary and apparent truths, and implicit assumptions
-borrow liberally from favored sources whence came particular views (for apologists, this is often and unfortunately WL Craig) such that even particular phrasing is based on particular sources
-mix mathematics, technical jargon, and even appeals to theorems in an attempt to bolster a position without understanding much (if any) of the mathematical or otherwise technical parts of "their" argument (most of the time because of they are "borrowing" from sources as described immediately above, and too often from WL Craig).
-etc.
This is a thread about arguments relating to infinite regress, which is in fact not a single type or class of "problems" but had to be seperated into different classes that were further divided by e.g., the type of analysis and purposes of different people critiquing or defending stances in a particular context or approach or even field! So, for example, there are those who refer to chicken/egg type problems in arguments as infinite regress, those who classify types of infinite regress arguments (or problems) into e.g., "vicious" vs. benign or even virtuous (for infinitism proponents), those whose approach is ontological in nature and deals more with Zeno-type arguments concerning infinities in space or processes in time, those whose bent is metaphysical and interested in e.g., causal relations among events and/or entities, and on and on.
On the other hand, this is a thread on an online forum. Exploring any of these in any detail is beyond what can probably be accomplished here, but we can try at least to be as clear about what we are saying, particularly when appealing to mathematics (which is supposed to be clear by virtue of being rigorous), and not conflating different approaches, different concepts, different aspects of different types of debates, etc.

When one says "infinity is not a number" then this could be a statement about one's personal belief based on one's personal philosophy of mathematics and associated metaphysics. Or it could be that one is adopting a particular number system and asserting that the number system has a set of elements (numbers) of which infinity isn't one. In the former case, there is no need to appeal to elementary calculus or make (false) claims about why some algebraic operation is or isn't possible, because neither such appeals nor such claims are relevant. One isn't making a mathematical claim that can be justified through trivial algebraic properties or limits. Alternatively, one IS making a claim about a number system, in which case it is clearly important to state why this is relevant at all. One can indeed use number systems in which negative and positive infinity are elements of the set of numbers and for which algebraic properties hold, and one can equally well consider number systems in which all kinds of familiar numbers don't exist. So what?
Mixing probability into this only further complicates matters when one doesn't specify the type of infinities one is using (even if only implicitly by e.g., specifying a measure, a density, and/or a mixed distribution that would rule out finite or countably infinite sample spaces). But it does provide a good example of how mixing one's intuitions about infinity with mathematical arguments can go terribly wrong is one is not sufficiently familiar with the mathematics. Same with physics, logic, etc.