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#41
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scientific proof and disproof
On 1/10/18 3:15 PM, Jonathan Thornburg [remove -animal to reply] wrote:
Phillip Helbig (undress to reply) wrote: Of course, technically, one can never prove the non-existence of something; On the contrary, one can indeed prove the non-existence of some things. For example, I am currently in a room which is approximately 2.5 meters by 5 meters by 2.5 meters in size. Given the known size of adult elephants, I can prove the non-existence of adult elephants in this room by looking around and not seeing any adult elephants. Abstracting a bit, for propositions X and Y, (a) if X implies Y, and (b) we observe not-Y, then (c) we have proven not-X. In the above example, X = "there is an adult elephant in this room" and Y = "I can see an adult elephant when I look around in this room". In the same way, one can prove the non-existence of (for example) hitherto-unknown Jupiter-mass planets orbiting within 10 astronomical units of the Sun: if such a planet or planets existed, they would cause substantial gravitational perturbations to the orbits of other planets. But we observe that there are no (unexplained) substantial gravitational perturbations to the orbits of the known planets in our solar system. Here we are taking X = "there is a hitherto-unknown Jupiter-mass planet orbiting within 10 astronomical units of the Sun" and Y = "there are substantial unexplained gravitational perturbations to the orbits of other (known) planets in our solar system". All of the above logic must consider Goedel's incompleteness theorem First Incompleteness Theorem from Wiki: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." (Raatikainen 2015) or perhaps another statement: all hypotheses cannot be considered true or untrue within any axiomatic structure defining the hypotheses. So, it is impossible to prove or disprove the complete universe structure based on any axiomatic structure Richard D Saam |
#42
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scientific proof and disproof
On 1/19/2018 7:19 AM, Richard D. Saam wrote:
On 1/10/18 3:15 PM, Jonathan Thornburg [remove -animal to reply] wrote: .. .. All of the above logic must consider Goedel's incompleteness theorem First Incompleteness Theorem from Wiki: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." (Raatikainen 2015) or perhaps another statement: all hypotheses cannot be considered true or untrue within any axiomatic structure defining the hypotheses. So, it is impossible to prove or disprove the complete universe structure based on any axiomatic structure Why do you say so? Goedel's incompleteness theorems only state that *some* statements are not provable. Statements you refer to here about the universe may not be of that type! As a matter of fact, Goedel's own incompleteness theorems *are* provable. (Using a formal system F of as mentioned, containing *some* statements are not provable. But incompleteness theorems still are!) -- Jos |
#43
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scientific proof and disproof
[Moderator's note: Followups, if any, should contain enough astronomy
content. -P.H.] On 1/19/18 1/19/18 1:19 AM, Richard D. Saam wrote: On 1/10/18 3:15 PM, Jonathan Thornburg [remove -animal to reply] wrote: [...] All of the above logic must consider Goedel's incompleteness theorem Not really (see below). First Incompleteness Theorem from Wiki: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." (Raatikainen 2015) Yes. That simply does not apply to what JT wrote. (And it misses some important conditions of the actual theorem.) or perhaps another statement: all hypotheses cannot be considered true or untrue within any axiomatic structure defining the hypotheses. This is NOWHERE CLOSE to Goedel's incompleteness theorems. Moreover, it is FALSE, at least in general. Perhaps you are thinking of Tarski's undefinability theorem -- I believe it is closer to what you seem to be trying to say. But it, too, does not apply to using math for physical models (where validity is not "true or false", but rather agreement with experiment). So, it is impossible to prove or disprove the complete universe structure based on any axiomatic structure You need to learn basic logic, as yours here is fatally flawed. You also need to learn the different between mathematics (incl. logic) and physics. No axiomatic structure has anything to do with the "structure of the universe". But, of course, axiomatic structures in the form of mathematics figure prominently in our MODELS of the universe. That the math is necessarily incomplete does not affect its applicability for constructing physical models. Tom Roberts |
#44
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scientific proof and disproof
On 1/19/18 11:00 AM, Jos Bergervoet wrote:
On 1/19/2018 7:19 AM, Richard D. Saam wrote: On 1/10/18 3:15 PM, Jonathan Thornburg [remove -animal to reply] wrote: .. .. All of the above logic must consider Goedel's incompleteness theorem First Incompleteness Theorem from Wiki: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." (Raatikainen 2015) or perhaps another statement: all hypotheses cannot be considered true or untrue within any axiomatic structure defining the hypotheses. So, it is impossible to prove or disprove the complete universe structure based on any axiomatic structure Why do you say so? Goedel's incompleteness theorems only state that *some* statements are not provable. Statements you refer to here about the universe may not be of that type! As a matter of fact, Goedel's own incompleteness theorems *are* provable. (Using a formal system F of as mentioned, containing *some* statements are not provable. But incompleteness theorems still are!) Taking the words from Goedel's Proof by Ernest Nagel and James R. Newman on their analysis of Goedel's 1931 paper: "If the Principia Mathematica is consistent, its consistency cannot be established by any meta-mathematical reasoning that can be mirrored within the Principia Mathematica itself. Meta-mathematical arguments establishing consistency of formal systems such as Principia Mathematica have been devised, but these proofs cannot be mirrored inside the systems that they concern." Goedel apparently tended towards a Platonic view by which objects had their own identity i.g. a triangle has its own identity independent of any logical consistency for it. Goedel had discussions in this regard with Einstein in the early 1950s probably in the context of the astrophysical universe. Surely Einstein had a much more deterministic view. Other discussions relate to the brain (or life in general) existing outside of logical consistency. Goedel would be in the affirmative. Richard D Saam [[Mod. note -- I think this discussion has now moved well outside the subject of this newsgroup ("research in astronomy/astrophysics"). Any further discussion should probably be over in the sci.math set of newsgroups. -- jt]] |
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