Godel and Boolean logic

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Vkothii

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propositional statements like: "if all people have colour vision, then they all see the same colours" are the same as {A and B} predicates; you need to assume that both statements in IF.. THEN clauses are a product, it really says IF A and B, then AB.

So the idea is of a completion; selecting or including A means you get AB, unless you only added A to B. (A or B) is inclusive or exclusive, which introduces the idea of negation, you have A and not B, or B and not A.

A plus B remain 'complete' with summation. They 'disappear' with propositions in propositional calculus and in Boolean logic (which is of course a propositional calculus).

Therefore the statement: "I am a liar" is a proposition which cannot be proven in any propositional formal logic, it can only be negated (inverted). Therefore it is a tautology that no formal logic system is complete.
 
Therefore we cannot use arithmetic to prove the universe is'nt lying, it might be.

We know it is about how much energy there is in QFT arithmetic, when we use that or QCD to measure relativistic scales. These theories are incomplete, as is the one we use to measure the relativistic scale.
So we're lying to ourselves, maybe as well.
 
Vkothii said:
Therefore we cannot use arithmetic to prove the universe is'nt lying, it might be.
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I'm not sure at all what you mean by this, or where you got the idea that one could use arithmetic to prove anything, let alone whether or not something is lying [let alone the notion that the universe could lie (??)].
 
I didn't get "the idea"; Godel got the idea.

We can't use the theories we have to prove that the universe isn't lying, we have a strong suspicion it is, when we use the 'best' theories we have.

All this can tell us, as Godel pointed out, is that our theories are incomplete; we can't tell.
We have our best interrogators on this universe's ass, and he hasn't cracked.
But we know he must be lying about something - the universe is a liar, ipso facto
 
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Vkothii said:
I didn't get "the idea"; Godel got the idea.
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Actually, that's incorrect.
Godel's incompleteness theorem was a result of a challenge presented by Hilbert.

Vkothii said:
We can't use the theories we have to prove that the universe isn't lying, we have a strong suspicion it is, when we use the 'best' theories we have.
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????

Who has this theory that the universe is 'lying'?
I've never heard of anyone stating such an anthropomorphosized notion.
 
But you can't prove that either 1) the universe is not lying (everything you see is an illusion, projected on a 3d screen, gravity is you accelerating through nothing, but you 'weigh' something, you have a direction through this space) 2) the notion that the universe is not lying is not anthopocentric.
 
The chapter after Godel is Turing, Markov and the advent of digital logic; in the machines we've built we ask questions of logical chains and loops (chains of loops and loops of chains), which are Markov processes or Turing tapes; they are also state-transition graphs and stack machines.

We can build a machine that produces a continuous answer or a (set of) discrete answer(s).
Logic itself is a machine that can build other machines - the universe seems to be constructed this way, because we are constructed the same way.

We understand some of the programming. We can ask: if we are constructed, what constructed us?
We have genetics, biochemistry and Evolution theories, these are the machines - we can ask them more questions and build more machines with them.

None of these "finished" machines we build can use a formal logic which is complete; obviously such machines are each finite, and cannot output more than a finite number of results; given a finite input and finite number of steps.

Therefore there cannot be any complete answer or question we can devise, or no machine we can finish building that will give either.

Therefore we will never be able to answer the question: is the universe a liar?
 
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Therefore the statement: "I am a liar" is a proposition which cannot be proven in any propositional formal logic, it can only be negated (inverted).
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Second order propositions can be handled formally in this notation , which is identical to Boolean algebra in all matters of overlap.

Note: the Wiki article inexplicably alternates between the notation of the original and the use of parentheses or brackets, thereby losing both the clarity of the notation (possibly the chief virtue of the entire approach) and any demonstration of the value of the approach. The proof of Leibniz's theorem given as a demonstration, for example, is far less persuasive than it would be in the original notation of the book.
 
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Are you saying you can prove: "no formal proposition is provable"?
Or I can use the notation to show the proposition cannot be proven, in a page or so?
 
vk said:
Are you saying you can prove: "no formal proposition is provable"?
Or I can use the notation to show the proposition cannot be proven, in a page or so?
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What? I can disprove that merely by reciting a valid syllogism.
 
You can disprove that no formal proposition is provable, or you can prove that you can't prove it isn't?
 
Well, here goes.
We can only construct well-formed formulas, and use them to construct iterative and recursive solutions within iterative/recursive domains; if there is a universal set P of problems, with a possible set S of solutions, then P is closed over S.
Any S which we use as a well-formed, logical or logically-physical proposition in P is also closed in S.

A proof is constructive, we apply a s from S, so that a p is constructed; but there is no proof of other than consistency.
What Godel called c-provable.
 
This from the wikii http://en.wikipedia.org/wiki/Godels_universe
By limiting oneself to sets defined only in terms of what has already been constructed, one ensures that the resulting sets will be constructed in a way that is independent of the peculiarities of the surrounding model of set theory and contained in any such model.

Define Def (X) = { {y | yεX and Φ(y,z1,...,zn) is true in (X,ε)} | Φ is a first order formula and z1,...,zn are elements of X}.

L is defined by transfinite recursion as follows:

* L0 = {}.
* Lα+1 = Def (Lα)
* If λ is a limit ordinal, then $$ L_{\lambda} = \bigcup_{\alpha < \lambda} L_{\alpha} \! $$.
* L = $$ \bigcup_{\alpha} L_{\alpha} \! $$ .

If z is an element of Lα, then z = {y|yεLα and yεz} ε Def (Lα) = Lα+1.

So Lα is a subset of Lα+1 which is a subset of the power set of Lα. Consequently, this is a tower of nested transitive sets. But L itself is a proper class.

The elements of L are called "constructible" sets; and L itself is the "constructible universe". The "axiom of constructibility", aka "V=L", says that every set (of V) is constructible, i.e. in L.
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Vkothii said:
But you can't prove that either 1) the universe is not lying (everything you see is an illusion, projected on a 3d screen, gravity is you accelerating through nothing, but you 'weigh' something, you have a direction through this space) 2) the notion that the universe is not lying is not anthopocentric.
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Of course. I completely agree (have understood for a long time).

I would say however, that the point is utterly moot.

Lying or not, things seem to be real. Sanity basically demands it.

So philosophically we can skirt this issue very very easily, but simply speaking of utility. Regardless of whether or not you believe a lie, that belief either served you or didn't as you see it. As such, your mind will judge its usefulness and as such establish its value to you.

Whether it is a lie or not is irrelevant if the truth of it can't be known.
 
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Regardless of whether or not you believe a lie, that believe either served you or didn't as you see it. As such, your mind will judge its usefulness and as such establish its value to you.
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But if "the universe is lying", and we have good reason to suspect that either we have theories that "lie" to us, or a universe that "lies" when we interrogate it with the theories, then it might be useful to investigate this apparent lie, or falsity of evidence.

Whether it is a lie or not is irrelevant if the truth of it can't be known.
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As Godel demonstrated, we can only know the truth of our ability to construct workable solutions, we can't know if these are universal solutions and we can't assume they will be able to answer any question, or that we can ever build such a machine.

However there is the Godel universe, which is 'in' the Von Neumann universe; the universe of logical machines that prove we can build, erm, more logical machines.
 
Vkothii said:
But you can't prove that either 1) the universe is not lying (everything you see is an illusion, projected on a 3d screen, gravity is you accelerating through nothing, but you 'weigh' something, you have a direction through this space)
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One doesn't need to prove the denial of an assertion to reasonably make that assertion. You're assuming that the universe operates exclusively via disjunction. The Law of Excluded Middle may very well not apply here.
(Although, I still have no idea what you mean by it 'lying')


Vkothii said:
2) the notion that the universe is not lying is not anthopocentric.
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How could it possibly not be such?
To lie necessarily implies an intentional account of obfuscation.
 
wesmorris said:
Of course. I completely agree (have understood for a long time).

I would say however, that the point is utterly moot.

Lying or not, things seem to be real. Sanity basically demands it.

So philosophically we can skirt this issue very very easily, but simply speaking of utility. Regardless of whether or not you believe a lie, that belief either served you or didn't as you see it. As such, your mind will judge its usefulness and as such establish its value to you.

Whether it is a lie or not is irrelevant if the truth of it can't be known.
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Well said wes.
Strictly speaking, no open system can be proved.
Thus, my first response in this thread.
Despite our inability to prove anything that we haven't created, we nonetheless must do things. And to do things, we must understand them (as best we can). And so, we move onward, doing quite well making use of pragmatically robust inductive reasoning, fully aware that while it's not certain (a la Hume), it works.
 
sherlok: welcome to the site, and the Philosophy forum in particular.

Firstly, do be so kind as to include the poster name when you quote them. It makes responding easier.

sherlok said:
You have to get the hang of what c-complete means; I think it means the following
Since you can't:

”

Means only that:

Implies: That from an anthropocentric propositional calculus, no question can be formed which can answer this question: "is the universe lying to us?"
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Do feel free to explain what you mean by any of the above.
 
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