Month: July 2010

The Axiom of Infinity as Easy Way Out?

I posed the question, Can one prove that there are infinite sets?  Researching this question, I consulted the text I studied when I took a course in set theory in a mathematics department quite a few years ago. The text is Karl Hrbacek and Thomas Jech, Introduction to Set Theory (Marcel Dekker, 1978). On pp. 53-54  we read:

It is useful to formulate Theorem 2.4 a little differently. We call a set A inductive if (a) 0 is an element of A; (b) if x is an element of A, then S(x) is an element of A. [The successor of a set  x is the set S(x) = x U {x}.]

In this terminology, Theorem 2. 4 is asserting that the set of natural numbers is inductive. There is only one difficulty with this reformulation: We have not yet proved that the set of all natural numbers exists. There is a good reason for it: It cannot be done, axioms adopted so far do not imply existence of infinite sets. Yet the possibility of collecting infinitely many objects into a single entity is the essence of set theory and the main reason for its usefulness in many branches of abstract mathematics.  We, therefore, extend our axiomatic system by adding to it the following axiom.

The Axiom of Infinity. An inductive set exists.

Intuitively, the set of all natural numbers is such a set.

Therefore, if we turn to the mathematicians for help in answering our question, we get the following. There are infinite (inductive) sets because we simply posit their existence! Thus their existence is not proven, but simply assumed. Philosophically, this leaves something to be desired. For it is not self-evident that there should be any infinite sets.  If there are infinite sets, then they are actually, not potentially, infinite.  (The notion of a potentially infinite mathematical set is senseless.)  And it is not self-evident that there are actual infinities.

I will be told that there is no necessity that an axiom be self-evident.  True: axiomhood does not require self-evidence.  But if an axiom is an arbitrary posit, then I am free to reject it.  Being a cantankerous philosopher, however, I demand a bit more from a decent axiom.  I suppose what I am hankering after is a compelling reason to accept the Axiom of Infinity.

A comparison with complex (imaginary) numbers occurs to me.  They are strange animals.  But however strange they are, there is a sort of argument for them in the fact that they 'work,' i.e. they find application in alternating current theory the implementation of which is in devices all around us. But can a similar argument be made for the denizens of Cantor's Paradise?  I don't know, but I have my doubts.  Nature is finite and so not countably infinite let alone uncountably infinite.  But caveat lector:  I am not a philosopher of mathematics; I merely play one in the blogosphere.  What you read here are jottings in an online notebook.  So read critically.

 


Nietzsche’s Definition of ‘Nihilist’

Der Wille zur Macht #585 (Kroener Ausgabe): 

Ein Nihilist ist der Mensch, welcher von der Welt, wie sie ist, urteilt, sie sollte nicht sein, und von der Welt, wie sie sein sollte, urteilt, sie existiert nicht.

A nihilist is one who judges of the world as it is, that it ought not be, and of the world as it ought to be, that it does not exist.

My translation is as beautiful as the German original.  Don't you agree?

Mel Gibson, Misplaced Moral Enthusiasm, and Real Threats

Mel Gibson is in the news again.  What I said about him on 1 August 2006 bears repeating:

What's worse: Driving while legally drunk at 87 miles per hour in a 45 mph zone, or making stupid anti-Semitic remarks? The former, obviously. And yet a big stink is being made about  Gibson's drunken rant. I call this misplaced moral enthusiasm.

Calling a Jew a bad name won't kill him, but running him over in your speeding 2006 Lexus LS 430 will. On the one hand, offensive words that no reasonable person could take seriously; on the other hand, a deed that could get people killed.

Here is what Gibson said: "The Jews are responsible for all the wars in the world," and, addressing the arresting officer, "Are you a Jew?" Now compare Gibson with Iran's Mahmoud Ahmadinejad who repeatedly has called for the destruction of the State of Israel. Ahmadinejad's is speech that incites unspeakable violence, unlike Gibson's drunken rant which is no threat to anyone. So let's forget about Gibson, and concentrate on real threats.

Necessitas Consequentiae versus Necessitas Consequentiis

Take the sentence, 'If I will die tomorrow, then I will die tomorrow.' This has the form If p, then p, where 'p' is a placeholder for a proposition. Any sentence of this form is not just true, but logically true, i.e., true in virtue of its logical form. Now every sentence true in virtue of its logical form is necessarily true. (The converse, however, does not hold: there are necessary truths that are not logically true.) Thus we can write, 'Necessarily(if p, then p)'  or

1. Nec (p –>p).

The parentheses show that the necessity attaches to the consequence, represented by the arrow, and not to the consequent, represented by the terminal 'p.' (When speaking of conditionals, logicians distinguish the antecedent from the consequent, or, trading Latin for Greek, the protasis from the apodosis.) Thus the above is an example of the necessitas consequentiae. This, however, must not be confused with the necessitas consequentiis, which is exemplified by

2. p–>Nec p.

In (2) the necessity attaches to the consequent. It should be obvious that (1) does not entail (2), equivalently, that (2) does not follow from (1). For example, although it is necessarily true that if I will die tomorrow, then I will die tomorrow, it does not follow, nor is it true, that if I will die tomorrow, then necessarily I will die tomorrow. Proving fatalism cannot be that easy. For even if I do die tomorrow, that will be at best a contingent occurrence, not something logically necessitated. (Think about it.)

To confuse (1) and (2) is to confuse the necessity of the consequence with the necessity of the consequent. This is an example of what logicians call a fallacy, i.e., a typical error in reasoning, and in particular a modal fallacy in that it deals with the (alethically) modal concepts of necessity and possibility and their cognates.

Class dismissed.

Can Federalism Save Us?

I fear that we are coming apart as a nation.   We are disagreeing about things we ought not be disagreeing about, such as the need to secure the borders.  The rifts are deep and nasty.  Polarization and demonization of the opponent are the order of the day.   Do you want more of this?  Then give government more say in your life.  The bigger the government, the more to fight over.  Do you want less?  Then support limited government and federalism.  A return to federalism may be a way to ease the tensions, not that I am sanguine about any solution. 

Federalism, roughly, is (i) a form of political organization in which governmental power is divided among a central government and various constituent governing entities such as states, counties, and cities; (ii) subject to the proviso that both the central and the constituent governments retain their separate identities and assigned duties. A government that is not a federation would allow for the central government to create and reorganize constituent governments at will and meddle in their affairs.  Federalism is implied by the Tenth Amendment to the U.S. Constitution: "The powers not delegated to the United States by the Constitution, nor prohibited to it by the States, are reserved to the States respectively, or to the people." 

Federalism would make for less contention, because people who support high taxes and liberal schemes could head for states like Massachusetts or California, while the  conservatively inclined who support gun rights and capital punishment could gravitate toward states like Texas. 

The fact of the matter is that we do not agree on a large number of divisive, passion-inspiring issues (abortion, gun rights, capital punishment, affirmative action, legal and illegal immigration, taxation, wealth redistribution, the purposes and limits, if any, of governmental power  . . .) and we will never agree on them.  These are not merely 'academic' issues since they directly affect the lives and livelihoods and liberties of people.  And they are not easily resolved because they are deeply rooted  in fundamental worldview differences.  When you violate a man's liberty, or mock his moral sense, or threaten to destroy his way of life, you are spoiling for a fight and you will get it.

Worldview differences in turn reflect differences  in values.  Now values are not like tastes.  Tastes cannot be reasonably discussed and disputed  while values can.  (De gustibus non est disputandum.) But value differences, though they can be fruitfully discussed,  cannot be objectively resolved because any attempted resolution will end up relying on higher-order value judgments.  There is no exit from the axiological circle.  We can articulate and defend our values and clarify our value differences.  What we cannot do is resolve our value differences to the satisfaction of all sincere, intelligent, and informed discussants. 

Consider religion.  Is it a value or not?  Conservatives, even those who are atheistic and irreligious, tend to view religion as a value, as conducive to human flourishing.  Liberals and leftists tend to view it as a disvalue, as something that impedes human flourishing.  The question is not whether religion, or rather some particular religion, is true.  Nor is  the question whether religion, or some particular religion, is rationally defensible.  The question is whether the teaching and learning and practice of a religion contributes to our well-being, not just as individuals, but in our relations with others.  For example,  would we be better off as a society if every vestige of religion were removed from the public square?  Does Bible study tend to make us better people?

For a conservative like Dennis Prager, the answer to both questions is obvious.  As I recall, he gives an example something like the following.  You are walking down the street in a bad part of town.  On one side of the street  people are leaving a Bible study class.  On the other side, a bunch of  Hells [sic] Angels are coming out of the PussyCat Lounge.  Which side of the street do you want to be on?  For a conservative the answer is obvious.  People who study the Bible with its Ten Commandments, etc. are less likely to mug or injure you than drunken bikers who have been getting in touch with their inner demons  for the last three hours.  But of course this little thought experiment won't cut any ice with a dedicated leftist.

I won't spell out the leftist response.  I will say only that you will enter a morass of consideration and counter-consideration that cannot be objectively adjudicated.

My thesis is that there can be no objective resolution, satisfactory to every sincere, intelligent, and well-informed discussant, of the question of the value of religion.  And this is a special case of a general thesis about the objective insolubility of value questions with respect to the  issues that most concern us.

Another  sort of value difference concerns not what we count as values, but how we weight  or prioritize them.  Presumably both conservatives and liberals value both liberty and security.  But they will differ bitterly over which trumps the other and in what circumstances.  Here too it is naive to  expect an objective resolution of the issue satisfactory to all participants, even those who meet the most stringent standards of moral probity, intellectual acuity, knowledgeability with respect to relevant empirical issues, etc.

Liberal and conservative, when locked in polemic, like to call each other stupid.  But of course intelligence or the lack thereof has nothing to do with the intractability of the debates.  The intractability is rooted in value differences about which consensus is impossible.  On the abortion question, for example, there is no empirical evidence that can resolve the dispute.  Empirical data from biology and other sciences are of course relevant to the correct formulation of the problem, but contribute nothing to its resolution.  Nor can reason whose organon  is logic resolve the dispute.  You would have to be as naive as Ayn Rand to think that Reason dictates a solution.

Recognizing these facts, we must ask ourselves: How can we keep from tearing each other apart literally or figuratively?  Guns, God, abortion, illegal immigration — these are issues that get the blood up.  I am floating the suggestion that federalism and severe limitations on the reach of the central government are what we need to lessen tensions. 

Example:  Suppose Roe v. Wade is overturned and the question of the legality of abortion is returned to the states.  Some states will make it legal, others illegal.  This would be a modest step in the direction of mitigating the tensions between the warring camps.  If abortion is a question for the states, then no federal monies could be allocated to the support of abortion.  People who want to live in abortion states can move there; people who don't can move to states in which abortion is illegal. Each can live with their own kind and avoid having their values and sensibilities disrespected.

I understand that my proposal will not be acceptable to either liberals or conservatives.  Both want to use the power of the central government to enforce what they consider right.  Both sides are convinced that they are right.  But of course they cannot both be right.  So how do they propose to heal the splits in the body politic?

Will Liberals Ever Retire the Race Card?

Why should they?  As good leftists, they believe the end justifies the means, and their shameless race-baiting is a means conducive to their ends.  It works.  That's why they do it.  They must at some level have an inkling of what vile people they are to employ such a  shabby tactic, but whatever sense of moral decency is left in them is quickly smothered by their lust to win at all costs.

And it is indeed a well-rehearsed tactic which amounts to collusion on the part of liberal-left journalists and others to smear conservatives.  This from the Christian Science Monitor:

When conservatives were criticizing Mr. Obama for his connection to the Rev. Jeremiah Wright in 2008, some JournoList members discussed a counterstrategy.

The Daily Caller writes that Spencer Ackerman, then of the Washington Independent, "urged his colleagues to deflect attention from Obama's relationship with Wright by changing the subject. Pick one of Obama's conservative critics, Mr. Ackerman wrote, 'Fred Barnes, Karl Rove, who cares – and call them racists.' "

See also Thomas Sowell's Race Card Fraud.  Excerpt:

The latest attack on the Tea Party movement, by Ben Jealous of the NAACP, has once again played the race card. Like the proverbial lawyer who knows his case is weak, he shouts louder.

This is not the first time that an organization with an honorable and historic mission has eventually degenerated into a tawdry racket. But that an organization like the NAACP, after years of fighting against genuine racism, should now be playing the game of race card fraud is especially painful to see.

You should also read the posts in my Race category. There is plenty of documentation there of the race-baiting and scumbaggery which are the now the marks of contemporary Democrats, liberals, and leftists.

And you should do your bit to push back.  The next time some scumbag of a liberal calls you a racist for standing up for fiscal responsibility or the rule of law, say this:  You lie about us, we'll tell the truth about you.

A Mosque Grows Near Brooklyn

Here.  Where is the money coming from?

Sarah Palin calls the building of this mosque an "unnecessary provocation."  As opposed to what, a necessary provocation?  But don't let Palin's infelicitous language distract you from the serious point she is making.  It is indeed  a  provocation, and the Islamists are testing us to see how far they can go and to see how weak and supine we are.  Will New Yorkers, sophisticated liberal fools that  many of them are, put up with this abomination a couple of blocks away from where their fellow citizens died horrible deaths because of a terrorist attack fueled by Islamist ideology?

The fact that the building of this mosque will be perceived as a provocation by a majority is sufficient reason to block its construction.  How can its construction do anything to improve relations between decent Muslims and the rest of us?

The first clause of the First Amendment of the U.S. Constitution guarantees the "free exercise" of religion.  True.  But is Islam a religion?  You will say, "of course!"  But perhaps you should be a bit more thoughtful.  Islam is a political ideology as much as it is a religion, and in this respect it is unlike Buddhism, or Christianity, or Judaism.  I have never heard any Jew call for the destruction of any Islamic state.  Muslims, however, routinely call for the destruction of the Jewish state.  When I lived in Turkey in the mid-nineties I was warned that preaching Christianity there could get one thrown in prison — not that I was about to do any such thing.  And Turkey in those days was a relatively 'enlightened' country compared to the rest of the non-Jewish Middle East. 

Muslims aren't very 'liberal,' are they?    They are intolerant in their attitudes and their behavior.  Now the touchstone of classical liberalism is toleration.  Toleration is good, but it has limits.  (See the posts in the category Toleration.)  So why should we tolerate them when they work to undermine our way of life?  The U. S. Constitution is not a suicide pact.  We are under no obligation to tolerate the intolerant. 

I said above that Islam is as much a political ideology as a religion. That is reflected in the fact that they have nothing like our church-state separation.  And please note that church-state separation has a good foundation in the New Testament at Matthew 22: 21: "Render therefore unto Caesar the things which are Caesar's, and to God the the things that are God's."  Please point me to the Koranic verse that enshrines the same idea.

Apologists say that Islam is a religion of peace.  Now 'peace' may be one of the meanings of Islam, but its dominant meaning is 'submission to the will of Allah as revealed to the propher Muhammad in the Koran.'  Let us also not forget that Muhammad was a warrior.  Was Jesus a warrior?  Buddha?  A religion founded by a warrior.  An interesting concept, that.  Somehow, I am more drawn to a religion whose founder says, "He who lives by the sword shall die by the sword."

So here is something to think about. The First Amendment guarantees freedom of religion.  But to apply the Amendment, one must raise and answer the logically prior question, What is a religion?    I rather doubt that the Founders had Islam in mind when they ensured the right to the free exercise of religion.  So we need to ask the question whether Islam counts as a religion in a sufficiently robust sense of the term to justify affording it full First Amendment protection.  To the extent that Muslims work to infiltrate and overturn our institutions and way of life, to the extent that they violate church-state separation, to the extent that they demand special privileges and refuse to assimilate, to that extent they remove  themselves from any right to First Amendment protection.

Addendum and Corrigendum (7/22)

I made a mistake in the last paragraph that I will now correct.  Although the sentence "I rather doubt that the Founders had Islam in mind when they ensured the right to the free exercise of religion" was true when I wrote it, expressing as it did a fact about my mental state, I now see that it is simply false that the Founders did not have Islam in mind.  See The Founding Fathers and Islam.  I thank Mark Whitten for the correction. 

But I do not retract my main point, which is that we ought to give careful thought to the question whether, as I put it above, "Islam counts as a religion in a sufficiently robust sense of the term to justify affording it full First Amendment protection. "  I am raising this as a question.  So-called liberals, however, being politically correct and therefore opposed to truly open discussion, will no doubt haul out their list of abusive epithets: racist, xenophobe, Islamophobe . . . .

I should point out that 'Islamophobe,' a term employed by the benighted Karen Armstrong, the renegade nun, is a particularly silly expression that only a liberal could love.  A phobia is an irrational fear.  If you use the word in any other way you are misusing it.  Fear of militant Islam is a rational fear.  One would hope that Armstrong, a Brit, would have a better grip on the English language.  These' -phobe' constructions are a dead giveaway that one is dealing with a PC liberal.  Take 'homophobia.'  Those who oppose homosexual practices neither fear it nor fear it irrationally.  Some have arguments against it.  In this case, then, the construction is doubly idiotic.  As for 'xenophobe,' that is a real word of English, but our benighted liberal pals seem not to know what it means.  It means 'irrational fear of foreigners.'  It does not mean 'someone who combats liberal-left nonsense.'  As one who has travelled the world and has lived for extended periods in Austria, Germany, and Turkey, I am hardly one who could be called a xenophobe.  Someone who opposes the infiltration of  his country by militant Muslims is not a xenophobe:  his fear is rational and it is directed not at Muslims qua Muslims but at Muslims qua militant subversives.

Over at Gnosis and Noesis, Professor Richard Hennessey rather pedantically and uncharitably picks at my "Muslims aren't very 'liberal,' are they?"  Do I mean that no Muslim is liberal?  Of course not.  A universal proposition can be refuted by a single counterexample.  (And it is worth noting en passant that a necessary universal proposition can be refuted by a single merely possible counterexample.)  Since it is obviously false that no Muslim is liberal, it is uncharitable to take my sentence as expressing that proposition.

One cannot assume that a sentence of the form Fs are Gs is always elliptical for a sentence of the form All Fs are Gs, or that a sentence of the form Fs are not Gs is always elliptical for a sentence of the form No Fs are Gs.  For example, 'Old people go to bed early' would not naturally be taken to mean that all old people go to bed early, which is plainly false, but that most do, or that old people tend to go to be early, or something similar.

Professor Hennessey seems also to be ignoring the context of my remarks, which is the construction of  mega-mosque near Ground Zero.  That, I submit, is an outrageous  provocation, a bit like building a Japanese  Shinto shrine in close proximity to the U.S.S. Arizona. (See here.)  I don't see how any rational person can fail to see that or fail to see that such a project cannot possibly bring together moderate Muslims and the rest of us.  And so it is reasonable to interpret the project as an initiative on the part of militant Muslims to take advantage of our tolerance and naivete in order to spread their religion and culture whose values are antithetic both to the Judeo-Christian tradition and to our Enlightenment values.

So that is the context in which a sentence like "They are intolerant in their attitudes and their behavior" is to be read.  The 'they' refers to militant Muslims: Muhammad Atta and the boys,  their enablers and supporters, those who flog and stone to death adulterers, those who would would impose Sharia, the clitorectomists, the Muslim fathers who murder their own daughters for adopting Western ways.  Our constitution forbids "cruel and unusual punishment."  Perhaps Hennessey can point me to the passage in the Koran that does the same.  And then there are Muslim taxicab drivers who refuse to pick up blind people with seeing eye dogs because of some lunatic Muslim aversion to dogs.  Others won't transport a person who has an alcoholic beverage in a closed container. That sort of fanaticism has no place in America.  I could go on, but the point is clear.

Just at the threat to the West in the 20th century was Communism, the threat to the West in the 21st is radical Islam. Both are totalitarian and internationalist.  Both are extremely skillful in recruiting young fanatic followers.  In one way the threat posed by militant Islam is far more dangerous than that posed by the Commies.  The Commies, being atheists and materialists, had a good reason not to deploy their nukes.  Muslims have no such reason.  (And it seems clear that they will soon be getting nukes thanks to Obama the Appeaser.) They are eager to move on to their crude paradise wherein they will disport endlessly with black-eyed virgins and get to wallow in the sensuousness that is forbidden them here.

For more on this delightful topic, see my Islamism category.

Another Strange Tale of the Superstitions

IMG_0310 The Superstition Mountains exert a strange fascination.  They attract misfits, oddballs, outcasts, outlaws, questers of various stripes, a philosopher or two, and a steady stream of  'Dutchman hunters,'  those who believe  in and search for the Lost Dutchman Gold Mine. This nonexistent object has lured many a man to his death.  More men than Alexius von Meinong's golden mountain, for sure.    Adolf Ruth, for example, back in the '30s.

Such appears to be the case once again this last week.  Three Utah prospectors, their brains addled by gold fever, entered this wild and unforgiving inferno of rocks and rattlesnakes  unprepared and appear to have the paid for their foolishness  with their lives.  Here is the story.

Or at least that is the story so far.  But there has to be more.  Why July when the temperature approaches 120 degrees Fahrenheit and the monsoon humidity adds a  further blanket of discomfort?  It is not as if they haven't been here before.  A couple of them were rescued last year.

And how do you get lost, if you are not totally stupid?  The central landmark of the entire wilderness is Weaver's Needle depicted in the first shot above.  It is visible from every direction, from the Western Sups to the Eastern Sups.  To orient yourself, all you have to do is climb up to where you can see it.  And then head for it.  To the immediate west and east of it are major trails that lead to major trailheads.

And why was no trace of them found despite  intensive searching with helicopters and dogs?  It is possible to fall into an abandoned mine shaft.  But all three at once?  Their plan, supposedly, was to search by day and sleep in a motel at night.  But then they wouldn't have gotten very deep into the wilderness and the chances of finding them dead or alive would have been pretty good.

IMG_0282 Maybe it was all a scam.  Maybe they never entered the wilderness at First Water.  They left their car there and hitchhiked out in an elaborate ruse to ditch their wives and families and their pasts.  But I speculate.  (If a philosopher can't speculate, who the hell can?)

I've hiked out of First Water many times, winter and summer.  I know a trail that you don't and is not on any maps that leads to Adolf Ruth's old camp at Willow Springs.  I've got half a mind to take a look-see . . .

On the Illicit Use of ‘By Definition’

What is wrong with the following sentence:  "Excellent health care is by definition redistributional"?  It is from a speech by Donald Berwick,  President Obama's nominee to head the Centers for Medicare and Medicaid Services, speaking to a British audience about why he favors government-run health care.

I have no objection to someone arguing that health care ought to be redistributional.  Argue away, and good luck! But I object strenuously to an argumentative procedure whereby one proves that X is Y by illict importation of the predicate Y into the definition of X.  But that is exactly what Berwick is doing.  Obviously, it is no part of the definition of 'health care' or 'excellent health care' that it should be redistributional.  Similarly, it is no part of the definition of 'illegal alien' that illegal aliens are Hispanic.  It is true that most of them are, but it does not fall out of the definition.

This is the sort of intellectual slovenliness (or is it mendacity?) that one finds not only in leftists but also in Randians like Leonard Peikoff.  In one place, he defines 'existence' in such a way that nothing supernatural exists, and then triumphantly 'proves' that God cannot exist! See here.

This has all the advantages of theft over honest toil as Russell remarked in a different connection.

One more example.  Bill Maher was arguing with Bill O'Reilly one night on The O'Reilly Factor.  O'Reilly came out against wealth redistribution via taxation, to which Maher responded in effect that that is just what taxation is.  The benighted Maher apparently believes that taxation by definition is redistributional.  Now that is plainly idiotic: there is nothing in the nature of taxation to require that it redistribute wealth.  Taxation is the coercive taking of monies from citizens in order to fund the functions of government.  One can of course argue for progressive taxation and wealth redistribution via taxation.  But those are further ideas not contained in the very notion of taxation.

Leftists are intellectual cheaters.  They will try to bamboozle you.  Listen carefully when they bandy about phrases like 'by definition.'  Don't let yourself be fooled.

"But are Berwick, Peikoff, and Maher really trying to fool people, or are they merely confused?"  I don't know and it doesn''t matter.  The main thing is not to be taken in by their linguistic sleight-of-hand whether intentional or unintentional.

 

Dissecting Leftism and Jihad Watch

Leftism and Islamism are the two main threats we face.  (Sorry, Al, your global warming is about as much a threat to us as your marital cooling.)  Both threats are totalitarian and  the threat is 'synergistic' inasmuch as leftists tolerate and enable militant Islam, which is obviously inimical to their modus vivendi, all the while displaying the most vicious intolerance of Christianity which is little or no threat to them.  I develop this theme in What Explains the Hard Left's Toleration of Militant Islam?

To help you think clearly about these important matters, I recommend Dissecting Leftism and Jihad Watch.

The Existence of Infinite Sets

A reader asked whether one can  prove that there are actually infinite sets.  Well, let's see.

It occurs to me that 'actually infinite set' is a pleonastic expresson. If there are infinite sets, then they are actually infinite, such that a potentially infinite set would be no set at all. For if there are mathematical (as opposed to commonsense) sets at all, then they are quite definite objects whose identity conditions are supplied by the Axiom of Extensionality: two sets are the same if and only they have all the same members. A mathematical set is not exhausted by its membership — it is not a mere plurality — since it is a one to their many; nevertheless, sets are rendered determinate by their members. (Let us for the moment not worry about singletons and the null set which give rise to their own difficulties.) 

It is worth noting that in Georg Cantor's oft-quoted definition, a set (Menge) is a collection of "definite and separate objects." (Contributions to the Founding of the Theory of Tranfinite Numbers, sec. 1) If the members of a set are definite and separate, then the same is true of the set itself. We could say that a math. set inherits its determinacy from the determinacy of its members. 

My point is that, if there are mathematical sets at all, then there is nothing potential, indeterminate, incomplete, or unfinished about them. Each such set is a definite single item distinct from each of its members and from  all of them.  It is a one-over-many. So if there are any infinite sets, then they are actually infinite sets, which is to say that talk of 'actually infinite sets' is redundant.

So our question becomes, Can one prove that there are infinite sets?

I don't know if one can prove it, but one can give an argument. (If a proof is a valid deductive argument the premises of which are self-evident, then damn little can be proven. In particular, the axioms of ZFC are far from self-evident, not that set theorists claim self-evidence for them. Is it self-evident that a null set exists?  Hardly.)

Here is an argument, where 'set' is short for mathematical (as opposed to commonsense) set.

1. There are sets.

2. There are infinitely many natural numbers: no finite cardinal is the number of natural numbers. Therefore,

3. If the natural numbers form a set, then they form an infinite set. (1, 2)

4. The natural numbers form a set.   Therefore,

5. The natural numbers form an infinite set. (3, 4) Therefore,

6. There exists an infinite set. (5)

This is a valid argument, and it renders reasonable its conclusion. But it does not prove its conclusion unless there are proofs for its controversial premises (1) and (4). I argued for (1) in Sets, Pluralities, and the Axiom of Pair.  But what is the argument for (4)?  Why must we think of the natural numbers as forming a set?