Uri Even-Chen wrote: > On 4/25/07, Shachar Shemesh <[EMAIL PROTECTED]> wrote: >> "Completeness" and "Consistency" relate to the relationship between the >> provability of an expression (syntax) and it's core truthfulness >> (semantics, or meaning). Since I was not talking about those, these >> hardly seem relevant. >> >> A theory cannot be either, because a theory is something that needs >> proof. In other words, using any moderately reasonable tools of proof, a >> theory can be correct and provable, correct and unprovable or incorrect >> (we usually do not let go of consistency because that leads to absurds). >> You will notice, however, that the theory is neither complete NOR >> consistent. These are measures not meant for theorems, but for logics. > > I agree. > >> Even for logics, the statement above is incorrect. Zero order logic is >> both consistent AND complete. > > I'm not sure what zero order logic is. How do you say "this sentence > is not true" in zero order logic? Zero order logic (propositional logic) has no relations. "this sentence is false" is represented as "!A", but as it has no relations, there is nothing that claims anything else about A beyond being false, which means it sees nothing special about you claiming that A is this sentence.
In other words, the logic is not granular enough to contain the paradox. > Good example. You assume this is true for all numbers. No, I do not. I can prove it's true for all number under the conditions specified. > Take any > positive number, Take any positive whole number. Read the premise correctly. > multiply it be 2, add one, devide by 4, and you get > either 1 or 3. Yes, you do. > Although I agree with you that it's true for any > number we can represent by a real computer, I don't *think* it's > infinitely true. See, the person doing assumptions is you. > I don't think integer numbers exist to infinity. It's your right, of course, but unless you have something substantial to back this up with, then I'm afraid any further discussion is based on differing opinions on how mathematics work, and are therefor meaningless. Out of curiosity, if natural numbers don't continue to infinity, there must be a maximal natural number, right? Assuming we call it "m", what is the result of "m+1"? > We > can define numbers so big, that 2n and 2n+1 is almost the same. Almost, yes. > In > any representation, whether in bits or in turing machines, if we > devide both numbers in 4 we will not necessarily get two different > results. See, not "any representation". The fact that you, or your computer, cannot solve a given problem does not impossible to solve make it. In mathematics, the numbers exist whether you can represent them in a finite space or not. If you have an infinite number of natural numbers, it is obvious you will need an unbounded number of bits to represent an unknown natural number. That does not make that number not exist. As a side note, a Turing machine has a semi-infinite storage, and would therefor have no problem to represent any natural number precisely. > I can't define such a specific number, since you will be > able to contradict me. That's where you prove yourself wrong. If a specific number you name turns out not to be the largest natural number, we have proven nothing. If, however, we are in agreement that ANY specific number you will name will not be maximal, or, in other words, that it is impossible for you to name the maximal number, then THERE IS NO MAXIMAL NUMBER. > It's an unknown unknown. Look what I wrote > about the largest known prime number. > > http://www.speedy.net/uri/blog/?p=25 I'm currently at a client's that employs content filtering. Your site is labeled as "propoganda" by fortinet. Being as it is that mirror.hamakor.org.il is labeled as "freeware download site", I wouldn't necessarily take their categorization too personally. Still, I cannot check your logic. > It's not a decision function. Decision functions return either 0 or > 1. I'm referring to the question whether there is any decision > function which can be proved not to be in O(the size of the input). I'm not sure, but as, like I said above, we do not speak the same language, it seems impossible to debate this in a meaningful way. Since your language also don't sit well with that of the rest of the mathematicians in the world, and seems not to be self consistent, then I'm not sure I will try hard enough. Shachar -- Shachar Shemesh Lingnu Open Source Consulting ltd. Have you backed up today's work? http://www.lingnu.com/backup.html ================================================================= To unsubscribe, send mail to [EMAIL PROTECTED] with the word "unsubscribe" in the message body, e.g., run the command echo unsubscribe | mail [EMAIL PROTECTED]
