Ontological argument criticism

[Ragna]

I want to criticize a version of the ontological argument which proposes atheism and does a reductio ad absurdum. I have also recently heard of "Gödel's ontological argument", but I'm not really into logic and I haven't found it with words so I can't criticize it yet. I would also like to know if the criticism I will do right now can work for Gödel's argument, and why / why not.

This is the version of the ontological argument I propose:

Axioms:

1. God is the most perfect (greatest) possible being.
2. A being which doesn't exist (possible) is less perfect than a being which exists (real).

Premises:

1. God doesn't exist in reality.
2. Therefore, there could be something greater than God: God existing in reality.
3. Therefore, there is something greater than the greatest possible being (contradiction).

Conclusion: God must as well exist in reality. (In more logical terms: If God is possible, he's necessary.)

If this version has something deeply different from Anselm's version then tell me so I can correct it.

Criticism:

It follows from axiom 2 that God being possible is less perfect than God existing in reality. Or, in another case, that they are both equally perfect possibilities.

Then, premise 2 is a non sequitur from premise 1. There could be something greater than the possibility of God, namely God existing in reality. This is non-contradictory, and it follows naturally from axiom 2.

3. Would read: "There is something greater than the possibility of the greatest possible being (The greatest being itself)."

There could be God existing in reality, but there could be not as well (just possible), that's why God is more perfect than his possibility. If God doesn't exist in reality, then there could be something more perfect than the possibility of God, not greater than God himself. There is no contradiction.

Aside, when I define God as the "greatest possible being" or "most perfect possible being", how perfect/great is that being? If I define my unicorn as "the equine with the longest horn", but it's impossible for equines to have horns, then the word "unicorn" would become a name for a horse whose horn measured 0 cm - the longest possible horn length in equines.

Questions:

1. Is my criticism right, if the argument is in fact a valid ontological argument version? (I hope)
2. Can this criticism work for Gödel's argument? Why / Why not? Can someone explain the differences in Gödel's argument without modal logic to me, in words?
3. Is it possible for a being like God to exist? What do we mean with this?

Definition of God: conscious entity that created the universe.

[EduChris]

...The paradox is that if the smallest integer that cannot be defined in less than eleven words does exist, it can be defined in less than eleven words...

What you are saying is this: "If X exists, then X does not exist." This an absurdity rather than a valid argument or even a paradox. In fact we already know that X doesn't exist for the very reason that any integer can be defined in under eleven words.

...Anyway, is this argument somewhat moot? Is there anyone who seriously supports any flavor of the ontological argument for the existence of God?

Godel's ontological theorem is not the same as Anselms. As far as I can tell, it has withstood all challenges on this thread.

[McCulloch]

any integer can be defined in under eleven words.

There are a finite number of English words. The Oxford English Dictionary, 2nd edition includes over 600,000 definitions.

Thus there are a finite number of phrases that contain less than eleven words. Someone who is current on enumerative combinatorics could even tell you just how many these are. Suffice it to say that this is a vast but finite number. The majority of sets of English words less than eleven words do not define any integer. So, the number of sets of English words with less than eleven terms that define a positive integer is certainly finite. A set of English words that define a positive integer can define one and only one positive integer. For example, the expression "An odd number greater than twelve" describes many integers and defines none.

There are infinite positive integers.

Thus there must be a positive integer that cannot be defined in under eleven words.

[EduChris]

...there must be a positive integer that cannot be defined in under eleven words...

Depends on how you define a "word." If you say that each integer has its own corresponding word, then there are at least as many words as there are integers. Dictionaries don't need to enumerate each of these words in order for them to be considered as words and used as words. Thus, 10000000000 is a word that defines its integer, and therefore any integer can be defined as "the smallest integer greater than its predecessor."

[McCulloch]

Thus, 10000000000 is a word that defines its integer, and therefore any integer can be defined as "the smallest integer greater than its predecessor."

10000000000 is not a word. It is a set of numerals representing an integer. In words, the integer it represents is ten billion.

No specific integer can be defined by the the phrase "the smallest integer greater than its predecessor" since that expression can apply to any integer.

Certainly any integer, M can be defined as "the smallest integer greater than {N}" where you insert an expression that represents the integer N in {N}. As long as N can be represented in less than six words, M can be represented in less than eleven.

1. There are a finite number of words composed of the letters a-z upper and lower case.
2. There are a finite number of sets of words with less than eleven elements that can be used to define a positive integer. (from 1 and elementary set theory)
3. There are an infinite number of positive integers.
4. Therefore, there must be some positive integers that cannot be defined with a set of less than eleven words. (from 2 and 3)
5. In any finite set of positive integers, there is one which is the smallest.
6. Therefore, there is a smallest positive integer not definable in less than eleven words. (from 4 and 5)
7. This number is defined by the expression "The smallest positive integer not definable in under eleven words"
8. Therefore there is no God.

[EduChris]

...10000000000 is not a word...

Opinion, not fact. But all we have to do is: 1) substitute the letters A through J for numerals 0 through 9, respectively; 2) have K represent the minus sign; and 3) add a paragraph explaining this in the dictionary and presto--we have an infinite number of words, no opinion necessary.

...Certainly any integer, M can be defined as "the smallest integer greater than {N}" where you insert an expression that represents the integer N in {N}...

My point exactly.

....As long as N can be represented in less than six words, M can be represented in less than eleven...

And N can be represented by precisely one word (see first paragraph, above).

[Ragna]

...10000000000 is not a word...

Opinion, not fact. But all we have to do is: 1) substitute the letters A through J for numerals 0 through 9, respectively; 2) have K represent the minus sign; and 3) add a paragraph explaining this in the dictionary and presto--we have an infinite number of words, no opinion necessary.

Your definition of word is erroneous.

The point in this is a word-idea correspondence. When you say definable under X words it means that it is definable using a finite number of concepts. How many character combinations you might want to do doesn't make them words. In fact, defining a word is much of an unresolved problem in linguistics, ask chinese people.

But 10000000000 is definable by adding the definable ideas of 2 and billion, so it constitutes two word-ideas, not 1, unless you are to defend the "fact" that 2 is half of a word .

Let's not get too much into linguistics, but I'm interested in philology as well and might be able to illustrate your fallacy here. Just to show how messy the concept of a word is, is "woodpecker" two words or one? In Spanish we say "pájaro carpintero" (carpenter bird) Just the space makes the difference? "Forgive" once included two words' meanings: "for" and "give", now... it's just one word because it has acquired one unseparable conceptual meaning. Unfortunately for your claim, numbers' meanings can be broken into lesser numbers' meanings. "Forgive", on the other hand, no longer corresponds to "for" and "give"'s meanings (semantic change).

Your example is analogous to saying that "two trees" is a word itself, a separate concept. Once you have "tree" and "two", you can have "two trees" by adding their meanings. It would be problematic if it weren't! Our minds have finite capacities (unless you are arguing that too), and if we needed X new conceptual informations for each time we numbered things, we would indeed have very little mathematical skills. Each time you do 15 x 2 through the algorithm you have been taught your mind is interpreting 15 as (10^1x1 + 10^0x5), that is, as 10 + 5. And 10 is itself only a combination of 1 and a power of 10. Aside, what about computers? Binary? All numeric information is reduced to two concepts - 0 and 1, in certain places. Infinite really? Please be more careful the next time you assert something as a fact and deem opinions unnecessary

[mgb]

Ok, if we are concerned with a pairing off of statements
and specific integers there is a rule which is as follows-

Rule A: "A particular statement can refer to one and only one integer"

Now consider the statement-

"Not definable in under eleven words"

This statement can potentially refer to and infinitude of
integers; the smallest integer satisfying this condition,
the next smallest and the next smallest and so on...
because once an integer satisfies this condition the
condition is transfered to the next integer and the next
and so on. This seems to contravene Rule A. I don't know
if this goes any way towards resolving the issue but I feel
that this word 'define' is very confusing because it is
really an existence argument rather than a full definition.

________________________________________________

Here's a bit from Wikipaedia-

The Berry paradox as formulated above arises because of systematic ambiguity in the word "definable". In other formulations of the Berry paradox, such as one that instead reads: "...not nameable in less..." the term "nameable" is also one that has this systematic ambiguity. Terms of this kind give rise to vicious circle fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal.[2] To resolve one of these paradoxes means to pinpoint exactly where our use of language went wrong and to provide restrictions on the use of language which may avoid them.

This family of paradoxes can be resolved by incorporating stratifications of meaning in language. Terms with systematic ambiguity may be written with subscripts denoting that one level of meaning is considered a higher priority than another in their interpretation. The number not nameable0 in less than eleven words' may be nameable1 in less than eleven words under this scheme.

[McCulloch]

For the benefit of EduChris, who seems to wish to debate the meaning of the word, word rather than delve into the implications of this paradox, I have restated the paradox.

1. There are a finite number of words composed of the letters a-z upper and lower case.
2. There are a finite number of sets of words with less than eleven elements that can be used to define a positive integer. (from 1 and elementary set theory)
3. There are an infinite number of positive integers.
4. Therefore, there must be some positive integers that cannot be defined with a set of less than eleven words. (from 2 and 3)
5. In any finite set of positive integers, there is one which is the smallest.
6. Therefore, there is a smallest positive integer not definable in less than eleven words. (from 4 and 5)
7. This number is defined by the expression "The smallest positive integer not definable in under eleven words"
8. Therefore there is no God.

Or if you really insist on being a stickler:

1. There are a finite number of words composed of the letters in the Latin alphabet upper and lower case.
2. There are a finite number of sets of words composed of the letters in the Latin alphabet upper and lower case with less than one hundred elements that can be used to define a positive integer. (from 1 and elementary set theory)
3. There are an infinite number of positive integers.
4. Therefore, there must be some positive integers that cannot be defined with a set of less than one hundred words composed of the letters in the Latin alphabet upper and lower case. (from 2 and 3)
5. In any finite set of positive integers, there is one which is the smallest.
6. Therefore, there is a smallest positive integer not definable in less than one hundred words composed of the letters in the Latin alphabet upper and lower case. (from 4 and 5)
7. This number is defined by the expression "The smallest positive integer not definable in under one hundred words composed of the letters in the Latin alphabet upper and lower case."
8. Therefore there is no God.