What makes an infinite set any more problematic than an empty set?

[PseuTonym]

The Roman numeral system doesn’t provide a symbol for zero, but surely the people who used that system had a word meaning something like “none.”

The question “How many horses do you own?” is difficult to answer directly if “zero” isn’t accepted as a number. However, one can say, “It is not true that there exists at least one horse that I own.” or “The set of horses that I own is equal to the empty set.”

Note: the question is directed to people who deny that an actual infinity can exist, and who insist on speaking of infinity as “potential.”

“Because Aristotle said so” isn’t a good answer, so please refrain from quoting anything that Aristotle wrote until after you have explained your own idea in your own words.

 

[Wesrock]

Thomas Aquinas did not deny that an actual infinite could exist (and Aristotlean-Thomism does not deny this either). Aquinas denied that an infinite regress in an essentially ordered series (per se) could exist. An infinite regress in an accidentally ordered series (per accidens) is perfectly plausible.

I suppose one way to think of an accidentally ordered series is the way a father begets a son. Whether or not any prior member of the series exists anymore (grandfather, great grandfather, etc…) has no bearing on the father’s power to beget a son. The father is capable of acting under his own power. Aristotle and Aquinas had no trouble with this type of series being infinite.

An illustration of an essentially ordered series is a steam train. The caboose is dependent on the power of the car in front of it for its motion, but the power of that car is derived from the power of the car in front of it, which is derived from the power of the car in front of it, which is derived from… you get the idea. The caboose is not dependent on just the one member in the series it is next to, it is dependent on ALL members in the series. But the power of motion all the cars has is derivative. It is not something natural to any of the cars. No car has the power to produce it’s own motion. No car has motion intrinsically as part of being a car. It’s not possible for any member in the series to give what it does not have. You cannot have an infinite regress where all members of the series have what they have detivatively – otherwise none would have that power. Therefore, there must be a member of the series (the engine car) which has the power of motion to give. A member whose power is not derivative from something else. Essentially, you cannot give what you do not have, and you cannot receive what can’t be given (because it’s not there). And this is a case where you are not dependent on just the previous member of the series, but the continuing action of all prior members.

Gaven Kerr presents a helpful way of formalizing this difference: accidentally ordered series can be represented as a series of one–one dependence relations where each member depends directly only on the previous member: (v→ w)→ (w→ x)→ (x→ y). In essentially ordered series, by contrast, the later members depend directly on (and derive their membership from) all the earlier members: (v→(w→(x→y))).

…

Many commentators, especially those who are not specialists in mediaeval philosophy, are tempted to read the first way as talking about a temporal succession of movers where each mover has been moved by some earlier mover at some point in time but whose current ability to act as a mover does not now depend on being moved by the earlier mover. When understood this way, Aquinas’s denial of the possibility of an infinite series seems, as Russell and Hick suggest, to involve a cognitive failure to grasp that an infinite series is, in fact, a possibility (cf. Russell, History of Western Philosophy, 453; Hick, Philosophy of Religion, 20). At best, Aquinas has some background argument against the infinite which he is assuming in his proof.

In fact, as we have seen, Aquinas is committed to the simultaneity of causation, so the series which are under discussion, even in the first way, are ordered in terms of ontological dependence, not temporal succession. Aquinas’s example of the stick moving the stone insofar as it is being moved by the hand is meant to serve as a paradigmatic instance of a sequence of simultaneous moved movers, a purpose for which Aquinas regularly cites it (In phys. VII 2.892; SCG II 38.13). He is considering an essential order of movers, not some accidental combination of various motions and movers that are not essentially related. If the hand were to cease moving the stick, the stick would cease to act as a mover pushing the stone. The moving activity that the stick performs is dependent on its being moved, on receiving from the hand the power to be a mover. Although Aquinas does think that these motions are simultaneous, the key issue here is ontological dependence, in this case the dependence of the activity of one thing on the activity of others. Even if there is some gap between the hand’s moving activity and that of the stick, as long as the continuing action of the stick depends on the continuing action of the hand, and the continuing action of the hand depends on the continuing action of the man and so on in infinitum, each mover will be a mover only insofar as it is being moved by all the previous movers.

tandfonline.com/doi/full/10.1080/09608788.2013.816934

Now, people who favor William Lane Craig and the kalam cosmological argument do say an actual infinite is impossible, but that’s neither here nor there for Aquinas, so I will not address that now.

Further reading:
edwardfeser.blogspot.com/2010/08/edwards-on-infinite-causal-series.html?m=1
edwardfeser.blogspot.com/2012/11/the-incompetent-hack.html?m=1
edwardfeser.blogspot.com/2015/06/cross-on-scotus-on-causal-series.html?m=1

 

[Vic_Taltrees_UK]

I am not familiar with the people you cite who draw a distinction between an infinity which “can exist” and one which “is potential”.

We possibly can only visualise approximations to infinity. I also think there are such things as more infinite and less infinite, also such a thing as more than infinite, and more than more than infinite.

In the “circumstances” I am convinced infinity is exceedingly exceedingly “probable” and is plenty plenty “potential”. This is “existence” in bucket loads!

Without interviewing them as to their concepts behind particular words, I wouldn’t understand why this wouldn’t satisfy them. I am “like wow” with it!

 

[Vic_Taltrees_UK]

Further to the interesting points brought up by Wesrock.

I think there is the less caused and more causing. And the less causing and more caused.

It would be stupid to think that such a large subject and field as “existence” wasn’t somewhat paradoxical. Paradoxes aren’t let-outs or get-outs. There is no logic without paradoxes.

Most languages have a word for “and”.

“A and B” is a good statement. “A therefore not B” is a faulty statement.

 

[Peter_J]

Language that is used with sets has lots of suggestive imagery. “Contains” “is in” “size” even the word “set” itself all point to a particular way of imagining sets.

 

[tm21]

Thomas Aquinas did not deny that an actual infinite could exist (and Aristotlean-Thomism does not deny this either). Aquinas denied that an infinite regress in an essentially ordered series (per se) could exist. An infinite regress in an accidentally ordered series (per accidens) is perfectly plausible. etc…

Excellent!

 

[yppop]

Although Cantor provided mathematicians with a method of dealing with the infinite, he did not resolve the problem of potential vs. actual infinity. Potential infinity is related to situations without limits, like strings of numbers that go on forever. Actual infinity represents arrangements that have limits, a minimum and a maximum, but has infinite members, like all the real numbers between 0 - 1. There is aleph1 real numbers in that range, an actual infinity. Strangely enough there are also aleph 0 rational in that same range. Both aleph 0 and aleph1 represent potential and actual infinities. Resolution of this potential/actual infinity allows us to deal more rationally with other dualities such as the physical and the abstract; discrete and the continuous; and the material and the spiritual.
yppop

 

[Peter_J]

Although Cantor provided mathematicians with a method of dealing with the infinite, he did not resolve the problem of potential vs. actual infinity. Potential infinity is related to situations without limits, like strings of numbers that go on forever. Actual infinity represents arrangements that have limits, a minimum and a maximum, but has infinite members, like all the real numbers between 0 - 1. There is aleph1 real numbers in that range, an actual infinity. Strangely enough there are also aleph 0 rational in that same range. Both aleph 0 and aleph1 represent potential and actual infinities. Resolution of this potential/actual infinity allows us to deal more rationally with other dualities such as the physical and the abstract; discrete and the continuous; and the material and the spiritual.
yppop

But is actual infinity anything more than a designation?

For example, if you ask what it means for a “set” to be infinite, the answer is that it “contains” infinity many elements. That terminology seems perfect natural, even unavoidable … but only because of the imagery invoked by the words “set” and “contains”.

Without that imagery, the “infinite set” terminology would be somewhat arbitrary, no?