Tuesday, March 31, 2009
infinity vs GODELS theorem
i was wondering if Gödel's incompleteness theorems are related to the infinte numbers in mathematics....... we can see that there are several unanswered quuestions because of the limitation mathematics........ besides, Can we think of any other ways to describe the existance and actions of our daily life? is mathematics based on numbers or describtion of things by the numbers?
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3 comments:
Binyam,
You have asked several questions all at once!
First of all, we need to get an idea about what Gödel's work was about.
Gödel's incompleteness theorems are concerned with axiomatic deductive systems - starting out with axioms and trying to deduce theorems. We developed a basic vocabulary for understanding these systems in last week's sessions.
What he showed was a bit similar to the situation we find with the claim "this sentence is false".
The claim "this sentence is false" cannot be true, because then it would be false.
But the claim "this sentence is false" cannot be false, because then it would be true.
This is just an easier-to-understand idea that gives us an inkling of what Gödel was talking about - so if we now shift from talking about true/false to provable/unprovable:
Gödel showed that there are mathematical systems (sets of axioms and the deductions made from them) that contain statements that are undecidable - that can never be proved true or proved false by the system itself.
This was a major blow to the hopes of some mathematicians, because it seems to place a unbreachable limit on certain types of mathematical knowledge.
Well, I think that, there actually are many other ways to explain the existence of our lives other than the use of mathematics. Is mathematics based on numbers or description of things by the numbers? Well, I would say that, mathematics is most often based on the description and explanation of how to get to something not just by numbers but other mathematical language.
For instance,the pythagoras theorem which my class raised issues on in class yesterday is a typical example. The theorm basically explains through mathematically understandable language how movement from one place to another in an area shaped like a right angled triangle is obtained. It explains the sysytematic link between distances, therefore describing and explaining how to obtain one length if you have the others. Even in calculus, the differncial symbols used, aid in describing how to obtain: for example the gradient of a line from the lines equation in an almost set and systematic order.
I really believe that, mathematics has its bases from real life situations and most often just tries to explain it, using its set and accepted axioms and theorems.
A slight digression...
Four brilliant mathematicians: Georg Cantor, Ludwig Boltzmann, Kurt Gödel, Alan Turing - what else have they got in common?
Do the research; give us the answer! When I was your age, it would take weeks to answer this question; but you have the tools to answer it in seconds!
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