How do most modern theologians, for example the established Roman Catholic Church, disprove the mathematical validity of the Burali Forti paradox?
The theological implications of the above paradox are obvious and apparent, and would seemingly pose the greatest and most profound threat to the stability of the present religious world, (i.e. as A.W. Moore explains the ‘totality of all sets’ (as conceived by Cantor) is logically equivalent to a null set).
And yet, this paradox has not resulted in the greatest religious controversy of our age nor in seemingly any profound theological debate, argumentation, or apologetics.
Why? And what have been the attempts of theological set theorists to create a mathematical refutation of this paradox?
How does my lack of a desire to participate in a debate within the forum of General Questions reflect newness or imply inexperience?
The Burali Forti paradox states:
“Now Omega* (and therefore also Omega) cannot be a consistent multiplicity. For if Omega* were consistent, then as a well-ordered set, a number D [delta] would belong to it which would be greater than all numbers of the system Omega; the number D, however, also belongs to the system Omega, because it comprises all numbers. Thus D would be greater than D, which is a contradiction. Thus the system Omega of all ordinal numbers is an inconsistent, absolutely infinite multiplicity.”
“This seems paradoxical, and is closely related Cesar Burali Forti’s “paradox” that there can be no greatest ordinal number… …As A.W. Moore notes, there can be no end to the process of set formation, and thus no such thing as the totality of all sets, or the set hierarchy. Any such totality would itself have to be a set, thus lying somewhere within the hierarchy and thus failing to contain every set.”
If God is equated with absolute infinitude, then the conclusion that there can logically be no totality of all sets (i.e. absolute infinitude) is potentially highly damning of religious orthodoxy.
What is the mathematical defense of religious orthodoxy against this argumentative proposition?
Why should we equate God with absolute infinitude? Why should we assume that God can be described by our mathematics? That’s no more obvious than a claim that God can be described by our physics. There’s no more reason to assume that we can know God’s cardinality than that we can know what material he’s made of.
i agree with wendell wagner expecting God to be bound by our mathmatics makes about as much sense as going up to linus torvolds, yelling cd usa/michigan/st joe county/st joe river/.secret fishing.spot . then conclude he had no part programing linux when he won’t take you there.
the programmer’s nature is not bound by the nature of the programs s/he writes.
> And what have been the attempts of theological set theorists
> to create a mathematical refutation of this paradox?
And there are, after all, so many theological set theorists. Just this summer I was at the W.C.T.S.T., the World Conference of Theological Set Theorists. It’s not really my main subject of interest, but I maintain a small interest in it. And since it was held in Washington, D.C. this year, just 10 miles from where I live, I decided I could attend this time. I couldn’t afford to get to last year’s conference in Prague, nor will I be going to next year’s in Buenos Aires.
The attendees filled the Convention Center in D.C., and those from out of town filled up half a dozen hotels. I suppose there were four or five thousand of us there. I presented the results of my work on how Aquinas’s Summa Theologica was flawed because he failed to take into account the Continuum Hypothesis.
I don’t recall any discussion of the theological implications of the Burali-Forti Paradox. I’ll tell you what though. I’ll submit a note for the Problems for Future Research section of the W.C.T.S.T. quarterly journal, Transfinite and Transcendent, and see if anyone has done any work on the problem.
Wow, I never realized there was such a large group of theological set theorists, semminarian-metamathematicians!!!
Thats incredible! I would love to read your article on Thomas Aquinas if you could post it somewhere on the internet! That’s really neat. How could I get involved with something like that?
The first statement does not seem to follow from the intial premise that we call the set of all numbers “omega”. Theres always another number thats bigger than D so the first statement is never true.
The trouble with infinity is that concepts are finite, whether they are mathematical or not. You can’t say an awful lot about something you can’t conceive of, whether you are a mathematical genius armed with the best logical tools or not.
Yeah, around here, too. Damn, I wish I could go to the donut shop for a peaceful double chocolate and black coffee without listening to their incessant yammering.
Translated to Theological terms, then it would roughly be “If God contains all things, then Goid must contain God, and then God must ne larger (greater, more encompassing) than God.” but that is false. God encompasses God by tautology, just as a quart of milk contains all amounts of milk up to and including a quart of milk, without contradiction.
The problem is that the mathematical paradox you cite depends on many specific premises that simply do not apply. Fr example there is NO theological principle that if God encompasses an entity X, there must be another entity Y that is greater than X (as is the case with the mathematical definition of Now Omega.)
Saying God is everywhere and sees everything is not the same as saying God is everything. The idea the God is within everything or “is everything” is contrary to most denominations of Christianity (and many other religions), and is a hallmark of many non-Christian religions (Paganism, Unitarianism, some Yogas, etc.) After all, if divinity is within you then you do not need Christ for salvation. You can do it yourself.
As for those religions that do believe God to be everything, others have already posted reasons why they would not care about mathematical paradoxes.
I’ve actually talked about something very similar to this. From Descartes’s Meditations on First Philosophy, V.5 (bolding mine):
He’s of course referring to Euclid’s Fifth Postulate here. For those who don’t know, Euclid’s Fifth was long accepted to be absolutely true (that is, that it followed from the other axioms and postulates), but for centuries mathematicians were unable to prove it. Descartes says, since it makes so much sense, it must be true; this is not, in his eyes, an assumption. The thing is, it’s not. Not necessarily. And if you don’t assume it’s true, you open up the realm of non-Euclidean geometry, which is quite revolutionary. This is pretty similar to non-Cantorian set theory, which is what you get if you don’t assume the Continuum Hypothesis is true.
So Descartes made a mistake, so what? The big deal is that his whole thesis was deriving the world from first principles, cogito ergo sum, not assuming anything at all. But then he goes and makes an assumption that is wrong! And not only is it wrong, but by not assuming it, you introduce a whole new interesting branch of mathematics.
Descartes might have been centuries ahead of his time if he’d just stuck with his principles.