> On 17 Dec 2018, at 08:50, Bruce Kellett <[email protected]> wrote: > > On Mon, Dec 17, 2018 at 5:59 PM Jason Resch <[email protected] > <mailto:[email protected]>> wrote: > On Mon, Dec 17, 2018 at 12:03 AM Bruce Kellett <[email protected] > <mailto:[email protected]>> wrote: > On Mon, Dec 17, 2018 at 4:30 PM Jason Resch <[email protected] > <mailto:[email protected]>> wrote: > On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett <[email protected] > <mailto:[email protected]>> wrote: > On Mon, Dec 17, 2018 at 1:50 PM Jason Resch <[email protected] > <mailto:[email protected]>> wrote: > On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett <[email protected] > <mailto:[email protected]>> wrote: > > Are you claiming that there is an objective arithmetical realm that is > independent of any set of axioms? > > Yes. This is partly why Gödel's result was so shocking, and so important. > > And our axiomatisations are attempts to provide a theory of this realm? In > which case any particular set of axioms might not be true of "real" > mathematics? > > It will be either incomplete or inconsistent. > > > Sorry, but that is silly. The realm of integers is completely defined by a > set of simple axioms -- there is no arithmetic "reality" beyond this. > > > The integers can be defined, but no axiomatic system can prove everything > that happens to be true about them. This fact is not commonly known and > appreciated outside of some esoteric branches of mathematics, but it is the > case. > > All that this means is that theorems do not encapsulate all "truth". > > Where does truth come from, if not the formalism of the axioms? > > You are equivocating on the notion of "truth". You seem to be claiming that > "truth" is encapsulated in the axioms, and yet the axioms and the given rules > of inference do not encapsulate all "truth". > > I think I worded that badly. What I mean is given that truth does not come > from axioms (since they cannot encapsulate all of it), then where does it > come from? Does it have an independent, uncaused, transcendent existence? > > I don't know what that would mean. I don't think the truth of arithmetical > statements comes from some underlying consistent model in which the axioms > are "true". How do you determine the truth of the Godel sentence in some > axiomatic system? Only by going to some more general system, not by reference > to some underlying model. > > > Do you agree that arithmetical truth has an existence independent of the > axiomatic system? > > Since truth does not equal 'theorem of the system', there is a sense in which > this is true. But it does not mean that the truth of any syntactically > correct statement is independent of any axiom set. > > > I agree that there are true statements in arithmetic that are not theorems in > any particular axiomatic system. This does not mean that arithmetic has an > existence beyond its definition in terms of some set of axioms. You cannot go > from "true" to "exists", where "exists" means something more than the > existential quantifier over some set. Confusing the existential quantifier > with an ontology is a common mistake among some classes of mathematicians. > > I agree, let us ignore "exists" for now as I think it is distracting from the > current question of whether "true statements are true" (independent of > thinking about them, defining them, uttering them, etc.). > > True statements are true by definition! > > > What I am curious to know is how how many of these statements you agree with: > > "2+2 = 4" was true: > 1. Before I was born > 2. Before humans formalized axioms and found a proof of it > 3. Before there were humans > 4. Before there was any conscious life in this universe > 5. As soon as there were 4 physical things to count > 6. Before the big bang / before there were 4 physical things > > "2+2=4" is a tautology, true because of the meanings of the terms involved. > So its truth is not independent of the formulation of the question and the > definition of the terms involved.
What about ExEyEz (x^3 + y^3 +z^3 = 33) ? Bruno > > Bruce > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
