User:Faustnh/Logic annotations provisional
On Gödel's Theorems.
Faustino Núñez Hernández
Jules Richard suggested what we can consider a good outline of Gödel's incompleteness theorems, about 30 years before Gödel [1].
Imagine a list of all arithmetic sentences.
Let's consider that each individual arithmetic sentence, gets associated to the set of numbers for which the arithmetic sentence is true. In other words, let's consider an association between every individual arithmetic sentence, and the set of numbers which that arithmetic sentence refers. This way, for example, the arithmetic sentence "all odd numbers" represents and gets associated to the set of all odd numbers.
Now let's create the "Gödel numbers" of all arithmetic sentences. We have imagined we have the list of all arithmetic sentences. Let's suppose we arrange or sort this list in an alphanumerical unambiguous order (for example, we use the ASCII sorting criteria), like in a dictionary. Now, we number every item or entry (arithmetic sentence) of this list, starting from the first arithmetic sentence: 1st item, 2nd item, 3rd item, and so on. This way, now every arithmetic sentence gets also associated to an own single respective number, according to its position on the sorted list.
Now let's consider another thing. Every arithmetic sentence is associated with two things: one, the set of numbers represented by the sentence; and two, the number assigned to the sentence according to its position in the list. Regarding this, we'll pay attention to that two possibilities can occur: either the position number assigned to a given arithmetic sentence for its position in the list, is also included as one of the numbers pertaining to the set of numbers represented by the sentence; or it doesn't happen. For instance, if the arithmetic sentence "all odd numbers" is given in the list the 20th position, then 20 is not included within the set of numbers referred by 20th arithmetic sentence, because 20 is an even number, not an odd number. But if, on the contrary, the arithmetic sentence "all odd numbers" would have been given the 21th position in the list, then the number 21 would positively be included within the set of numbers represented by the 21th arithmetic sentence, because 21 is an odd number.
From this point, we'll build a new arithmetic sentence that is correct and legitimate, and that consequently should be also incorporated into our list of all arithmetic sentences; but that, however, cannot be incorporated into the list:
"The set of all positioning numbers (used for numbering the sentences of the list), that are not included inside the respective sets of numbers, represented by their respective numbered sentences".
For example, if positional number 20 was assigned to the arithmetic sentence "all odd numbers", then number 20 would not be included inside the set of numbers represented by its own arithmetic sentence, because 20th arithmetic sentence represents the set of all odd numbers, but 20 is even. So, 20 would have to be included into the set of numbers represented by the new arithmetic sentence we've just built in the previous paragraph.
The question now is: why this new arithmetic sentence we have built, couldn't be incorporated into our comprehensive list of all arithmetic sentences?.
Let's imagine we could incorporate our new arithmetic sentence into our list of arithmetic sentences. Then, our new arithmetic sentence should be assigned a positional number in the list: let's refer this number as "r".
The critical question is: is "r" included inside the set of numbers represented by our new arithmetic sentence?.
If it is, then our new arithmetic sentence says it shouldn't be.
But if it isn't, then our new arithmetic sentence says it should be.
( The apparent solution to this difficulty seems to be to include our newly created arithmetic sentence into another separate parallel list of sentences , defined with respect to the first list , so that no " critical question about a number r " within the first list can be formulated ; but this solution won't make impossible that the same problem can be reproduced again regarding the new built parallel list of sentences ).
On contradictions and paradoxes.
In conditions of declared or understood uniqueness of existence, something can't exist more than one time in declared or understood different inter-excluding forms.
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A certain object " B " can exist according to the existence of another different object " A ", in the following form:
If Object-A = something , then Object-B = another thing
If Object-A = 2nd something , then Object-B = 2nd another thing
etcetera...
... but object " B " existing according to itself, that is:
If Object-B = something , then Object-B = another thing
If Object-B = 2nd something , then Object-B = 2nd another thing
etcetera...
... doesn't make sense, in conditions of declared or understood uniqueness of existence of object " B " ( for example, in conditions of declared or understood initialization, start, originary creation, or birth of object " B " ).
( When an object " B " is going to be originary or initially created, in the beginning , object " B " does not exist ; there exist other different objects than " B " , that are used for building " B " , without " B " existing ; then " B " is created and starts existing , from objects different from it ; so object " B " cannot take part or participate in its own originary or initial creation ).
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