FOM: Goedel: truth and misinterpretations

[V. Sazonov]

Torkel Franzen wrote:


as I study and ponder various interesting
> results in mathematical logic, I'm led to reflect e.g. that
> 
>    (1) Even if ZFC is in fact consistent, it seems that most likely
>        no argument proving this to everybody's satisfaction will
>        ever be found.


I do not think that science should care about everybody's 
satisfaction. It has different goals - knowledge, even not 
convincing anybody. As to ZFC and even PA or PRA or even 
simpler formalisms I have no idea what could be a proof of 
their consistency (in the real sense of this word - no human 
being, even no computer will find a contradiction). We can 
only say that no contradiction have been found yet. But who 
knows? 

The physical law f = m*a also was experimentally true until 
Einsten came. Could you imagine the proof of f = m*a or of  
any other physical law "to everybody's satisfaction"? 

I even do not know, does it mean that we both agree concerning (1)? 
That is consistency of ZFC probably holds as a physical law, 
but, as any phisical law it cannot be proved in any sense, 
except by confirming it by experiments (or by deriving from 
other laws) and probably by some intuition. 

However, strictly speaking something stops me from asserting 
a deep analogy with physical laws. Formal proofs in ZFC among 
all other physically possible seqences of symbols (actually 
arising in our world either randomly, or with the help of 
human brains) appear too seldom. The consystency of ZFC will 
be usually confirmed by the trivial reason that the currently 
considered sequence is not a proof at all. Newertheless, some 
analogy exists. Here the main point is confirming by some 
intuition. All of us have some intuition concerning ZFC. 

> 
>   Now, we need not here consider whether the reflection (1) is *justified*,
> for I am told by you and V.Kanovei - who here, I really think, adopt
> a highly sophisticated philosophical approach 


quite normal and very simple, from my point of view! 


as opposed to my naive
> way of thinking - that it doesn't even make sense. So *why* doesn't it make
> sense? It seems to make good sense to me.
> 
>   An answer might be: it makes no sense because the infinite realm of
> derivations from ZFC is illusory. 


It depends on how you understand this world. As physicians 
in the case of f = m*a or something quite differently? 
Do you consider also imaginary proofs of non-feasible lengh 
(say 2^1000) or not? (If not, what does it mean?) But in your 
previous analogous statements you mentioned mathematical TRUTH 
(independent on provability). In that cases I have no idea how 
to interpret you. 


And then a natural response might
> be: who cares? Illusory or not, the question remains whether it is
> theoretically possible to derive a contradiction from the axioms of
> ZFC. To assume that it is not - i.e. that ZFC is in fact consistent -
> is to make an assumption about a matter of objective fact.

Physical law about sequences of symbols written, say, 
in hard disks of contemporary or even future computers, 
the law like f = m*a? Or in some higher sense? 

> 
>   At this point the exchange can take several routes. The view that
> there is no such matter of objective fact has its exponents, and as
> I've said before, I think Wittgenstein is the most interesting of
> these. 


I do not know what do you mean. 


But however the argument goes, it is a separate question
> whether it is even possible to do logic, mathematics, computer science,
> and so on, without making the kind of use of mathematical statements
> exemplified by (1).


Will you confirm, please, that we both understand this (1) 
(which is strongly different from the former (1) about TRUTH) 
in the same "physical" way, as I explained above. 


Note, that THIS DOES NOT MEAN THAT I THINK IN ANY PHYSICAL 
OR OBJECTIVE WAY ON THE UNIVERSE OF SETS DESCRIBED BY ZFC. 
ONLY AS ILLUSION! BUT PROOFS ARE CONSIDERED AS PHYSICAL OBJECTS. 


Vladimir Sazonov