We’re deep inside contingency and necessity this week. And one good joke. Regular followers of this series will recall St Thomas was not immune of the temptation to pun. Chapter 83 may be skimmed.
Chapter 82 Objections Against The Statement That God Wills Not Of Necessity Things Other Than Himself, In That It Involves Impossibilities (alternate translation)
 …Likewise neither does it follow that there is changeableness. For if there is no potentiality in God’s will, the reason why, in His effects, He does not of necessity give preference to the one alternative, is not because He is considered to be indifferent to either alternative, so as to be at first potentially willing either, and afterwards willing actually (whereas He is always actually willing whatsoever He wills, with regard not only to Himself but also His effects); but it is because the thing willed is not necessarily related to the divine goodness, which is the proper object of the divine will; in the same way in which we say that an enunciation is not necessary but possible where the predicate is not necessarily related to the subject.
Hence when we say: God wills this effect, this statement is clearly not necessary but possible, in the same way as a thing is said to be possible, not in reference to a potentiality, but because it is neither necessary nor impossible for it to be, as the Philosopher teaches (6 Metaph.). Thus the statement that a triangle has two equal sides is possible, yet not in reference to a potentiality, since in mathematics there is neither potentiality nor movement. Therefore the exclusion of the aforesaid necessity does not remove the unchangeableness of the divine will…
Notes As always, recall that God is outside time. God creates time. This is why the discussion of unchangeableness. We must relate to God, who is changeless and outside time, from being inside of it. Time is change. We cannot therefore ascribe to God’s mind powers that only belong to us. These few chapters thus answer the seeming objections that “God changes Him mind” or “God is or was surprised” or “God is open to the future” and such like.
Chapter 83 That God Wills Something Other Than Himself By A Necessity Of Supposition (alternate translation)
 …For it has been proved that the divine will is unchangeable. Now that which is once in an unchangeable thing cannot afterwards not be therein: since we say that a thing is changed when its condition is different now to what it was before. Therefore, if God’s will is unchangeable, supposing that He will something, it is necessary by supposition that He will it.
 Again. Everything eternal is necessary. Now that God will some particular effect to exist is eternal: for His willing, like His being, is measured by eternity. Therefore it is necessary. Not however if we consider it absolutely: because God’s will has not a necessary relation to this particular thing willed. Therefore it is necessary by supposition.
 Further. Whatsoever God could do, He can do, for His power is not diminished, as neither is His essence. But He cannot now not will what He is supposed to have willed, since His will is unchangeable. Therefore He never could not will whatever He has willed. Therefore it is necessary by supposition that He willed, as also that He will, whatever He willed: neither however is necessary absolutely, but possible in the aforesaid manner.
Notes Say this one three times fast! Okay, so this is true, but unsatisfying in some sense, and will be, too, because it doesn’t tell us why God does what He does nor how. But then we aren’t God and our finite minds cannot comprehend such things anyway. Think about Job!
 Moreover. Whosoever wills a thing, necessarily wills those things which are necessarily requisite to that thing, unless there be a defect on his part, either through ignorance, or because he is led astray from the right choice of means to the end in view, by some passion. But these things cannot be said of God. Wherefore if God, in willing Himself, wills something other than Himself, it is necessary for Him to will all that is necessarily required for what is willed by Him: even so is it necessary for God to will that there be a rational soul, supposing that He wills a man to be.
Notes And so He did will.
Chapter 84 That God’s Will Is Not Of Things Impossible In Themselves (alternate translation)
 HENCE it is clear that God’s will cannot be of things that are impossible in themselves.
 For the like are those which imply a contradiction in themselves: for instance that a man be an ass, which implies that rational is irrational. Now that which is incompatible with a thing excludes some of those things which are required for that thing: for instance, to be an ass excludes man’s reason. If, then, He wills necessarily the things that are required for those He is supposed to will, it is impossible that He will those that are incompatible with them. Hence it is impossible for Him to will things that are simply impossible.
Notes Ha ha ha! Anyway, here’s the answer to the unstoppable cannon ball meeting the immovable flag pole or any other man-made contradiction such as God can will 2 + 2 = 5. God cannot will the impossible. And this isn’t a limitation, but a perfection. This, really, is the reason the universe is predictable and comprehensible. The next arguments amplify this. Incidentally, how many readers made it to the joke?
 Again. As was proved above, God, by willing His own being, which is His own goodness, wills all things as bearing a likeness to Him. Now in so far as a thing is incompatible with the notion of being as such, it cannot retain a likeness to the first, that is, the divine being, which is the source of being. Wherefore God cannot will that which is incompatible with the notion of being as such. Now just as irrationality is incompatible with the notion of man as such, so is it incompatible with the notion of being as such, that anything be at the same time a being and a non-being. Hence God cannot will affirmation and negation to be true at the same time. Yet this is implied in everything which is in itself impossible, that it is incompatible with itself, in as much as it implies a contradiction. Therefore God’s will cannot be of things impossible in themselves…
Notes Hence mathematics! And don’t think about going all Godel here. That man’s theorems are about proofs, and proofs are the collection of necessarily true premises that demonstrate the truth of a proposition. Things can be true without us proving them. What Godel showed was that men cannot prove everything without assuming; indeed, this was always plain. Axioms, initial premises, are assumptions. And these axioms, given by a certain form of induction (I use this word in classical and not modern sense), must be in a sense, gifts of God.
Categories: Philosophy, SAMT
Gödel proved that numerical theories of a certain power are unprovable. Using that part of number theory that is provable, otherwise his proof would not work.
In other words, you are able to proof that 2+2=5, and because of that God is not able to cause 2+2 to be 5.
Now, Gödel says that certain statements in number theory are unprovable. But if God is unable to cause unprovable number theoretic results, the inability of God to cause them is actually the proof that those statements are true statements. Which means that it can be proven that they are true.
Which is a contradiction.
Sander, that also proves that God is unprovable, but simply an axiom that has valid alternatives that don’t need the God-axiom.
“What Godel showed was that men cannot prove everything without assuming; indeed, this was always plain.”
I think you’re trying to refer to the incompleteness theorems, which of course are not this at all. What Godel showed was not “always plain”, but turned the world of mathematics on its head overnight. Everyone since Euclid understood that you need to state assumptions to prove anything. Do you really think that this is all that Godel showed? There are popular accounts of Godel’s results all over the place. Sure you can find one that’s comprehensible to you.
” Incidentally, how many readers made it to the joke?”
I don’t get it.
Essentially: There are true statements in a discourse which are not provable in that discourse.
“Essentially: There are true statements in a discourse which are not provable in that discourse.”
No, there are statements which status is “don’t know.”
Or more exact: “not ever knowable”
No, there are statements which status is “don’t know.” [unknowable]
This begs the question. It assumes “knowable” and “provable” are the same, and the same as “true.”
Since everyone is talking about Godel, I had a question for the more learned. Given system A with its axioms and methods, there are questions in A that cannot be proved in A. These question may be proved by some other system B with its own axioms and methods.
Either the axioms of A and B are consistent or they are not. If they are consistent, then there is some system C that contains axioms of both A and B. This system C has questions it can raise but cannot answer. If they are not consistent, i.e. some axioms are contradictory, then we have no reason to believe that the answers given in B for questions from A are true.
Therefore, since A and B are arbitrary and potentially infinite in the axioms, there are always questions that cannot be answered from any formal system.
Does this work or do I misunderstand Godel’s theorem?
“there are always questions that cannot be answered from any formal system.”
You can always add one additional axiom that makes the undecidable proposition true (or false). It’s clear what that is, I hope.
In fact it is not clear, unless you simply make the proposition axiomatic, but that does not solve the problem I am proposing, because you will only end up with further questions and axioms ad infinitum.
Sorry, I must not understand what you are getting at.
“Given system A with its axioms and methods, there are questions in A that cannot be proved in A.”
This is false as there are theories — non-trivial ones — that are decidable and complete. What the (first) Göedel theorem says is that any theory T extending or interpreting Peano arithmetic (one can get by with less than Peano Arithmetic, but nevermind) is arithmetically incomplete, that is, there is one arithmetical statement P such that T neither proves P nor not-P (*). Two points (1) the arithmetical qualifier is crucial as it is a hardly interesting observation that for a theory T there are statements in the language of T that T cannot decide (2) the proof is constructive that is, given a T there is an algorithm that spits out the desired P.
(*) sometimes this is paraphrased by saying that there is an arithmetical P that is not provable in T and yet it is true. But the “is true” here can be misleading because it is not the usual mathematical sense, e.g. true in all models of T.
“These question may be proved by some other system B with its own axioms and methods.”
Yes, but this is trivial, for if T is consistent and P is independent of T then P is (trivially) provable in the consistent T + P.
“Either the axioms of A and B are consistent or they are not. If they are consistent, then there is some system C that contains axioms of both A and B. This system C has questions it can raise but cannot answer. If they are not consistent, i.e. some axioms are contradictory, then we have no reason to believe that the answers given in B for questions from A are true.”
I do not understand what the reasoning is supposed to be here.
“Therefore, since A and B are arbitrary and potentially infinite in the axioms, there are always questions that cannot be answered from any formal system.”
The second sentence does not follow from the first, so I am not sure what you are getting at. At any rate if what you mean is that there is no way at all to consistently extend Peano arithmetic to a theory T that decides every arithmetical statement then yes (subject to technical caveats like T remaining recursively axiomatisable). And?
Godel …. Hmmm
G. Rodrigues, Nu heb ik tijdens mijn studie geofysica heel wat wiskunde gehad, maar jouw relaas gaat volledig over mijn hoofd heen, en ik vermoed met mij het overgrote deel van de reaguurders op dit blog. Is je reactie wellicht bedoeld om de anderen te imponeren met je wiskundige welbespraaktheid? Welnu, dat is je gelukt, dus zou je het nu ook willen uitleggen aan lezers die geen universitaire graad in de wiskunde hebben? Dan hebben we namelijk iets van een dialoog. Dank je.
Rodrigues’s answers are probably as simple as can be in the context of Godel.