There might be a hair to split between what is conceived versus what is realized or actual. But, I would approach it as God must be slightly less than infinity and greater than everything else. However uncountable infinity would move the scale in a way that frustrates that conjecture.

Anselm's ontological argument fails! — TheMadFool

Yeah, it fails but, first and foremost, because it is, at most, merely valid and not sound, only the idea of God (essence) is 'demonstrated' but not the existence of the idea's referent as "the proof" also sets out to do. And, as an 'a priori argument', the OA (Proslogion) is only 'true by definition', thus vacuous with respect to a posteriori facts of the matter. So again, my friend, I think your "potential infinity vs actual infinity" argument misses the forest for the trees (& what said :smirk: ).

Note the underline bit, "...larger than any real or natural number". Sounds remarkably similar to "...no greater can be conceived" the main premise in the ontological argument. — TheMadFool

Didn't read any more of the thread than this, only jumping in with a mathematical correction. In math there are quantities that are larger than any finite number, but way smaller than other transfinite numbers. For example the cardinal $\aleph_0$ is greater than every finite number, yet smaller than $\aleph_1$, which is smaller than $\aleph_{\text{googolplex}}$. I realize this is primarily a religious thread and not a mathematical one so forgive the interruption. But your statement is no longer "remarkably similar" to "no greater can be conceived," as of the 1870's. Georg Cantor was the first human to conceive of, and logically prove the mathematical existence of, an endless hierarchy of larger and larger infinities. Infinity, at least mathematical infinity, is no longer just one single thing that's "larger than any finite number." It's an entire complex world of interrelated ideas of ordinal and cardinal infinities, respectively expressing order and quantity.

Many humans, mostly math majors, easily conceive of these things once they've had the proper training. The theologians should pay attention. Cantor himself was a very religious man, and believed that after his endless hierarchy of infinities, the ultimate infinity was God. He called it the Absolute infinite, and denoted it $\Omega$. Cantor's mathematics is universally accepted now, while his theological ideas are forgotten by everyone except historians. I imagine he'd be disappointed by that.

Just sayin'. I'll leave you learned theologians alone now, I don't know anything about the subject. Well I know one bit, when I think of the ontological argument I know about William Lane Craig and his misuse of set theory to prove the existence of the Christian God. So I haven't got a very good impression of theological arguments involving infinity.

There might be a hair to split between what is conceived versus what is realized or actual. But, I would approach it as God must be slightly less than infinity and greater than everything else. However uncountable infinity would move the scale in a way that frustrates that conjecture. — Cheshire

If God is less than anything he can't be "that than which nothing greater can be conceived." Infinity is the numerical representation of that than which nothing greater can be conceived (God). Ergo, since actual infinities don't exist, God too can't exist.

Thank you for your comment but, read my reply to Cheshire above. Cantor believed that God = infinity and actual infinities exist - he was right about the first but wrong about the second. Together these two (one true, the other false) constitute the premises of my argument that God can't exist!

Yeah, it fails but, first and foremost, because it is, at most, merely valid and not sound, only the idea of God (essence) is 'demonstrated' but not the existence of the idea's referent as "the proof" also sets out to do. And, as an 'a priori argument', the OA (Proslogion) is only 'true by definition', thus vacuous with respect to a posteriori facts of the matter. So again, my friend, I think your "potential infinity vs actual infinity" argument misses the forest for the trees (& what ↪fishfry said :smirk: ). — 180 Proof

Yes, the a priori - a posteriori distinction is important but do you know that theoretical physics (a priori) decides what kind of experiments (a posteriori) should be conducted. Also, many particles were first predicted by a priori physics theories and then, intriguingly, confirmed through experimentation!

No :sweat: ... read Popper or Peirce, my friend. No scientific entity is 'true by definition'.

No :sweat: ... read Popper or Peirce, my friend. No scientific entity is 'true by definition'. — 180 Proof

Look, I ain't the brightest bulb on the chandelier but I know one thing for a fact - physics is slowly metamorphosing into a full-fledged branch of mathematics. In math, some things are true by definition!

Note the underline bit, "...larger than any real or natural number". Sounds remarkably similar to "...no greater can be conceived" the main premise in the ontological argument. — TheMadFool

I don't think that Anselm was defining God as being "larger than any real or natural number." That would be a category error.

When Anselm talks about "a being than which no greater can be conceived" by "greater" he means something like "better" or "more awesome."

I don't think that Anselm was defining God as being "larger than any real or natural number." That would be a category error.

When Anselm talks about "a being than which no greater can be conceived" by "greater" he means something like "better" or "more awesome." — Michael

Mathematical infinity is the numerical representation of "that than which nothing greater can be conceived." What I've attempted to do is mathematize Anselm's conception of God. It's basically translating Anselm's God into a number, that's all. For instance, I say God is green, then mathematically God is a wavelength of 555 nm. Mathematizing Anselm's god is not just a trick-shot, it's germane to the ontological argument as the word "greater" in "...that than which nothing greater can be conceived" is inherently quantitative. It appears people failed to connect the dots, missed what was right under their noses.

Mathematical infinity is the numerical representation of "that than which nothing greater can be conceived." — TheMadFool

Cardinality notwithstanding, mathematical infinity is the numerical representation of that number than which none larger can be conceived. But that's not what Anselm is talking about. You're just equivocating.

When Anselm talks of being greater he's likely considering such things as power, intelligence, benevolence, etc., and likely includes such properties as having created the universe or being the moral authority, and so it makes no sense to try to mathematize his concept.

Anselm's God isn't just the mathematician's notion of infinity. They're two distinct concepts.

Mathematizing Anselm's god is not just a trick-shot, it's germane to the ontological argument as the word "greater" in "...that than which nothing greater can be conceived" is inherently quantitative.

It's not just quantitive. Jane is a greater piano player than Jim. Jim is a greater friend than Sam. These aren't things that can be assigned some numerical value.

greater — Michael

is quantitative. If you don't believe me, in math "greater"' is symbolized as ">". Nothing more need be said.

Jane is a greater piano player than Jim. Jim is a greater friend than Sam. — Michael

I can, given some leeway, easily mathematize that in terms of,

1. The length of time a person spends listening to either Jane's or Jim's piano pieces.

or

2. How large the audience is each of their performances

or

3. How many times their performances have been viewed on youtube
.
.
.

so on and so forth

is quantitative. If you don't believe me, in math "greater"' is symbolized as ">". Nothing more need be said. — TheMadFool

Anselm isn't using the mathematician's notion of "greater than" when he uses the phrase. You're equivocating.

I can, given some leeway, easily mathematize that in terms of,

1. The length of time a person spends listening to either Jane's or Jim's piano pieces.

or

2. How large the audience is each of their performances

or

3. How many times their performances have been viewed on youtube
.
.
.

so on and so forth

None of these are a measure of a person's piano-playing ability.

premises of my argument that God can't exist! — TheMadFool

Maybe God put all these ideas in your head. I can never understand the mindset of people who use logic to talk about the existence of God. God exists outside of the bounds of logic. God is a matter of faith, not logic. One believes or not. Or one has experienced miracles or not. But of course we all experience miracles every day, being alive, breathing the air. All the better if we have a roof over our head and we know where our next meal is coming from. If God didn't provide that, I'm a damned lucky fool. I don't believe in an anthropomorphised God, but I don't believe in an entirely mindless universe either. I'm an agnostic: "a person who believes that nothing is known or can be known of the existence or nature of God or of anything beyond material phenomena; a person who claims neither faith nor disbelief in God." That fits me to a T.

..no greater can be conceived." — TheMadFool

From where I'm sitting, it isn't necessary to invoke mathematics. As several folks have pointed out, the expression "greater than" must be based on some clear definition of "greatness" - and this definition must include some means/mechanism for comparing and deciding which of two "beings" is "greater than" than the other.

The ontological argument fails in this regard.

There are also many other equally valid reasons for rejecting the ontological argument that are mentioned in the 4th paragraph of the article you linked. I hope I'm not violating any forum rules by quoting:

Just as the ontological argument has been popular, a number of criticisms and objections have also been mounted. Its first critic would be Gaunilo of Marmoutiers, a contemporary of Anselm's. Gaunilo, suggesting that the ontological argument could be used to prove the existence of anything, uses the analogy of a perfect island. Such would be the first of many parodies, all of which attempted to show the absurd consequences of the ontological argument. Later, Thomas Aquinas rejected the argument on the basis that humans cannot know God's nature. David Hume also offered an empirical objection, criticising its lack of evidential reasoning and rejecting the idea that anything can exist necessarily. Immanuel Kant's critique was based on what he saw as the false premise that existence is a predicate, arguing that "existing" adds nothing (including perfection) to the essence of a being. Thus, a "supremely perfect" being can be conceived not to exist. Finally, philosophers such as C. D. Broad dismissed the coherence of a maximally great being, proposing that some attributes of greatness are incompatible with others, rendering "maximally great being" incoherent.

Anselm isn't using the mathematician's notion of "greater than" when he uses the phrase. You're equivocating.

I can, given some leeway, easily mathematize that in terms of,

1. The length of time a person spends listening to either Jane's or Jim's piano pieces.

or

2. How large the audience is each of their performances

or

3. How many times their performances have been viewed on youtube
.
.
.

so on and so forth

None of these are a measure of a person's piano-playing ability. — Michael

I would be equivocating iff I use two different meanings of the word "greater". I'm not. The word "greater" is inherently quantitative. It even has its own mathematical symbol ">". All I did was take Anselm's "greater" and rendered it in mathematical terms. Since Anselm's God is "that than which nothing greater [...]", mathematically, that's infinity. So, if Anselm believes God exists, his claim boils down to one about the existence of an actual infinity which is not as cut-and-dried as we/I would have hoped.

As an analogy regarding the quantitative/mathematical nature of the word "greater", I'd like to mention an issue that's very sensitive for people who are obese - weight. If I say Tom's weight is greater than Eric's what I actually mean is Tom's weight, say 90 kg, > Eric's weight, 70 kg. I can't make it clearer than that I'm afraid.

From where I'm sitting, it isn't necessary to invoke mathematics — EricH

See my reply to Michael above!

Thanks for the excerpt from the Wiki page about the ontological argument. I'm currently focused on my own refutation though. My argument differs from the rest because Anselm's argument is wholly predicated on the meaning of "greater" in "that than which nothing greater [...]"


To those interested

To tell you the truth though, this entire discussion is ultimately about the existence of actual infinities!

To tell you the truth though, this entire discussion is ultimately about the existence of actual infinities! — TheMadFool

And what exactly does this mean? Any and every line segment is an actual infinity of points - each point having an address that can be written down.

And what exactly does this mean? Any and every line segment is an actual infinity of points - each point having an address that can be written down. — tim wood

I thought of it that way too but if the set of points in a line segment is a completed infinity, it should be possible to make a list of them and...wait for it...the list should have a last entry. I'm afraid that's not possible. Actual infinities? :chin:

It's an entire complex world of interrelated ideas of ordinal and cardinal infinities, respectively expressing order and quantity. — fishfry

Attempt at a coherent question here. Maybe best to leave it simple. What is an infinite ordinal?

If God is less than anything he can't be "that than which nothing greater can be conceived." Infinity is the numerical representation of that than which nothing greater can be conceived (God). Ergo, since actual infinities don't exist, God too can't exist. — TheMadFool

How do you go about conceiving infinity? If infinity doesn't exist(never conceived to completion), then God can be less than infinity and exist.

However, actual infinity is highly controversial.

Actual infinities don't exist. — TheMadFool

If they are controversial, how are you justified in asserting, without supporting reasoning, that they don't exist? Or, conversely, if they don't exist, how can they be controversial?

Yeah, it fails but, first and foremost, because it is, at most, merely valid and not sound, o — 180 Proof

Relative to your belief system, please share what would make it sound (if you can) ?

Attempt at a coherent question here. Maybe best to leave it simple. What is an infinite ordinal? — tim wood

Terrific question. I'm working on a response. I can make it simple but I can't make it short. Anything I write is going to be grossly off-topic to this thread. How do the assembled multitudes feel about that? I'll just go ahead and write up my response and post it here later or tomorrow, whenever it gets done. As far as being on topic, as I mentioned, Cantor thought that the ultimate ordinal was God. And maybe it is.

How do you go about conceiving infinity? — Cheshire

Take a class in set theory. Or just contemplate the set of natural numbers {0, 1, 2, 3, ...} that you learned about in grade school. Everybody believes in the natural numbers. So why would it be difficult to conceive infinity? We did it in grade school. "Name the biggest number." "A zillionty-zillion." "A zillionty-zillion plus one!" That's infinity. The sequence keeps going forever.

And the transfinite ordinals are what you get when you count through all the natural numbers and keep on going. I'll explain that in detail soon. Perhaps it should be in its own thread. I didn't realize we were in the ontological argument thread, this is definitely going to be a little off topic.

So why would it be difficult to conceive infinity? We did it in grade school. — fishfry

Did we really though? I think we conceived the conditions for an infinity. I can conceive 10s or maybe hundreds and infer about millions and billions, but saying I'm thinking about the impossible whole of infinity seems reaching.

I don't feed trolls, lil troll.

Potential infinity is a matter that's settled. However, actual infinity is highly controversial.

My refutation of Anselm's argument:

Actual infinities don't exist.

I believe I may have found a problem with one of your premises though.

"Actual infinities don't exist," doesn't seem to follow from their being controversial.

Did we really though? I think we conceived the conditions for an infinity. I can conceive 10s or maybe hundreds and infer about millions and billions, but saying I'm thinking about the impossible whole of infinity seems reaching. — Cheshire

You just need to stretch your imagination. Think of each of 0, 1, 2, 3, 4, ... where you can always "add 1." Then think of all of these together in one place, what we call the set of natural numbers {0, 1, 2, 3, 4, 5, ...}.

Now I admit that this is not a very natural thing to conceive of. But if you spend enough time studying it, it does become second nature and you can conceive of it perfectly well. I tell you honestly that I have no problem conceiving of the set of natural numbers, because I spent enough time in school studying it.

"Conceive of" is a very weak standard of existence because it's subjective. What one person can't conceive of, another person finds commonplace.

It may be true that you personally can't conceive of the set of natural numbers, but I assure you that people who study math in college end up with a very clear picture of it in their minds. And far larger sets too.

Think about the real number line that they taught you in high school. It has points at all the integers like ---, -4, -3, -2, -1, 0, 1, 2, 3, 4, ..., going to infinity in both directions. Between each whole number are the rationals like 1/2 and 2/3 and 47/99, etc., which are three rational numbers between 0 and 1. And there are all the irrationals like sqrt(2) and pi and e and so forth. Each tiny little dimensionless point on the line is the location of some real number; and there are a whole lot of those, provably more of them than there are integers.

And you learned this in high school. So you learned it, but never really spent much time conceiving it.

My thesis is that if you had simply spent a few years studying the real line and its properties, you'd end up conceiving of it perfectly well.

Now I agree that this is far short of conceiving of the infinity of God or the infinity of the cosmos, or whatever. Mathematical infinity is a very limited kind of infinity, it's the one that we can talk about using symbolic logic. But still, it's infinity and lots of people do conceive of it.

Yeah, it fails but, first and foremost, because it is, at most, merely valid and not sound, o
— 180 Proof

Relative to your belief system, please share what would make it sound (if you can) ?
1hOptions — 3017amen

TMF, I realize that it's your OP (and a good one), if 180 can't answer that question, I think it may mean that his particular belief system is in question.

I'm not sure he really understands the concept behind the ontological argument. But be patient, we'll see how he responds (if he does at all). And so at this point, I wouldn't take too much stock in his analysis (considering he generally struggles with his 'premises' from other threads).

Anyway, in layman's terms, unfortunately he's all bark and no bite. Just wanted to point that out; didn't want you to be misled... .

Anyway, in layman's terms, unfortunately he's all bark and no bite. — 3017amen

And you're just a vicious little coward. You boasted about what you would accomplish in a debate, and then ran away it, just as you run away from everything of substance. I have asked you myself many questions that you never responded to.

Pathetically, it was only my "bark" that scared off that "vicious" little girl. :smirk:

I believe I may have found a problem with one of your premises though.

"Actual infinities don't exist," doesn't seem to follow from their being controversial. — Count Timothy von Icarus

Good point! Your question takes me back a coupla weeks to this thread :point: Proving A Negative/Burden Of Proof.

Related stuff:

1. Argumentum Ad Ignorantiam

2. Pascal's Wager

Am I justified in inferring the nonexistence of actual infinities based on the fact that no actual infinities have been demonstrated to exist?

My argument would look like this:

1. There are actual infnities OR There are no actual infinities

2. We couldn't prove that There are actual infinities.

Ergo,

3. There are no actual infinities

The argument form employed is the disjunctive syllogism. The crucial question: [We couldn't prove that] There are actual infinities = There are no actual infinities???

When does the absence of proof of a proposition P imply the falsehood of P?

Imagine there's a book that contains every proof that's possible. A proof is what transform uncertainty (p v ~p) for a proposition p into certainty ( p, if not ~p).

Suppose now I go through this book from cover to cover and I find no proof for the proposition, There are actual infinities. In other words the proposition can't be proven which simply means the proposition (There are no actual infinities) can't be true (no proofs exist that demonstrate P).

If the proposition can't be true but it has to be true or false, it follows that the proposition must be assigned the truth value false i.e. There are no actual infinities!

However, the situation I'm in is different. The book of proofs I mentioned earlier isn't something I possess and so a handful of people failing to prove There are no actual infinities falls short of a thorough scan of the book of proofs. Thus the fallacy, argumentum ad ignorantiam.

However, if someone put a gun to my temple and say, "make the most logical choice given the following...or else...bang! bang!"

1. There are actual infinities or There are no actual infinities

2. Your opponent in the debate hasn't been able to prove There are actual infinities

My reasoning would proceed like this:

My opponent hasn't proved There are actual infinities. Insofar as my opponent and I are concerned, There are actual infinities can't be true [think of it like having gone through a certain number of pages in the book of proofs]. Since There are actual infinities hast to be either true or false, the most logical option is There are actual infinities is false i.e. There are no actual infinities.

There are no actual infinities. — TheMadFool

The transcendental number pi is encoded in and manifest (instantiated) by every non-abstract, or concrete, circular object which makes it an uncountable actual infinity, no?