By Godel's incompleteness theorem it is impossible to prove consistency of the current widely accepted foundation of mathematics ZFC within ZFC. But this theorem says nothing about existence or non-existence of a possible formal proof for inconsistency of ZFC within ZFC that means it is possible that some day set theorists or other working mathematicians find an inconsistency between two mathematical theorems.
My first question is about any possible option which could be chosen by set theorists, logicians and mathematicians in this imaginary situation.
Another question is about possible impacts of discovering an inconsistency in mathematics on philosophy of mathematics and some fields of human knowledge like theoretical physics which use mathematics extensively.
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It seems weakening the current axiomatic foundation of mathematics in any sense (including removing a particular axiom or moving to another weaker axiomatic system) causes an expected problem. In fact avoiding the contradiction by means of weakening our axiomatic system (which seems the only accessible choice) sends some accepted parts of current mathematics into the realm of "non-mathematics". Thus in this case we need to choose between different parts of mathematics that which one is good and useful and which one is not. This could be the matter of many discussions. For example if the Axiom of Choice (AC) is a part of that contradiction then by removing it from the foundation we will lose many useful tools of mathematics including many essential theorems like "Every vector space has a basis" that harms linear algebra extensively.
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Most textbooks on set theory comment on this possibility and the majority oppinion seems to be that if such inconsistencies would be discovered somewhere, it would probably be not too difficult to device some technical modifications (there are many modified axiomatic set theories in use already) that would avoid these inconsistencies (at the risk to introduce new ones, this would be a job guarantee for foundational mathematicians).
Most textbooks on set theory comment on this possibility and the majority oppinion seems to be that if such inconsistencies would be discovered somewhere, it would probably be not too difficult to device some technical modifications (there are many modified axiomatic set theories in use already) that would avoid these inconsistencies (at the risk to introduce new ones, this would be a job guarantee for foundational mathematicians).
That weakening the present axiomatic system seems the only accessible choice does not rule out the choice of instead strengthening mathematics, as Gödel's theorem, apparently often overlooked, also suggests, but true such a meaningful pursuit would be at the risk of ruling out many job guarantees.
I just ended a proof that ZFC set theory is relative consistent. The proof is available as an answer to the question "is the set theory consistent?".
Suppose that we discover a contradiction in ZFC or in mathematics in general. Even in this event we can go ahead by considering e.g. paraconsistent logic, see e.g http://en.wikipedia.org/wiki/Paraconsistent_logic, and Carnielli, Walter; Coniglio, Marcelo E. and Marcos, J, (2007). "Logics of Formal Inconsistency". In In D. Gabbay and F. Guenthner (eds.). Handbook of Philosophical Logic, Volume 14 (2nd ed.). The Netherlands: Kluwer Academic Publishers. pp. 1–93. ISBN 1-4020-6323-7.
Furthermore, the existence of a contradiction seem to show a direction for Many-valued logic, Category Theory and dialectics, etc. In these frameworks we may study ZFC as an internal model, and we may add up with a better view for mathematics. All these problems are related to two-valued logic. Fuzzy Logic e.g. does not suffer from such things.
NF (New Foundations) [W. Quine, 1937] has some variants which are consistent, e.g. NFU (NF with urelements) [R. Jensen. 1969]. By now we can use ZFC set theory as no paradoxes emerged, and if things go wrong with ZFC, we can replace it by its alternative, i.e. NFU.
Mathematics is not inconsistent, and then nothing catastrophic happens to mathematics, since ZFC set theory is consistent and thus mathematics is consistent, see answer to the question "is the set theory (ZFC) consistent?".
Can we use the “brackets-placing rule"(１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．) applied in the divergent proof of Harmonious Series (1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．) to produce infinite numbers bigger than 1/2 or 1 or 2 or 3 or 4…?
What you against is not my idea. Please see following proof:
1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．． （1）
=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．． （２）
>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．． （3）
=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity （4）
Such an antique proof (given by Oresme in about 1360), though very elementary, can still be found in many current higher mathematical books written in all kinds of languages. And this proof has been acting as a basic theory of mathematics producing many mathematical conclusions.
But the problem is we meet a modern version of “Achilles and Tortoise” Zeno’s Paradox------- Harmonious Series Paradox.
This kind of “infinite-infinitesimals paradox” tells us:
1, in Harmonic Series, we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite infinitesimals in Harmonic Series by “brackets-placing rule" to change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity;
2, the “brackets-placing rule" to get 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items in Harmonic Series corresponds to different runners with different speed in Zeno’s Paradox while the items in Harmonic Series corresponds to those steps of the tortoise in Zeno’s Paradox. So, not matter what kind of runner (even a runner with the speed of modern jet plane) held the race with the tortoise he will never catch up with it.
In this case of “how many number items bigger than 1/2 can we have by ‘brackets-placing rule’ in harmonic series”, either finite or infinite is “illogical” in present science.
Actually our operation in Harmonious Series Paradox is: to change an infinitely decreasing Harmonious Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity--------this means that not matter what kind of runner (even a runner with the speed of modern jet plane) held the race with the tortoise in Zeno’s Paradox, the runner will never catch up with it.
“Achilles and Tortoise” Zeno’s Paradox has disclosed the "real infinite--potential infinite" fundamental defects in present traditional infinite related science theory system. In this theory system, the critical defects are unsolvable because we are not allowed to forget either "real infinite" or "potential infinite" in such paradox (exactly same situation happen in Harmonious Series Paradox).
That is why this paradox family has been troubling us human for more than 2500 years.
As a researcher, I am sincerely looking for the different opinions to improve the work, would you please be so kind as to tell me where I do wrong, welcome to join the discussion. The foundation of infinite related area in our science is really very week. Through discussions we can get better understanding on the topic, any ideas are devoted to our scientific work.
Sincerely yours!
Geng
Dear Akira,
I agree with you. All these problems and paradoxes points to the correct direction! This is the great service of paradoxes and foundational problems. I disagree with the habit of mathematicians to factor out the "devil", by proving that belong to a set of measure zero etc. I believe that we should analyse paradoxes and foundational problems to find a way out!
Dear Mr. Costas Drossos, Happy New Year!
We can always get enlightening ideas from you, thank you.
Sincerely yours,
Geng
Could we have a short break for following inconsistent dish?
When we study ”the meaning of zero" and the location of zero in “number spectrum” in our mathematics, an unbalanced defect can be easily discovered: “zero" appears on one side of the “number spectrum” as a kind of mathematical language telling people a situation of “ nothing, not-being,…”; but on the other side of the “number spectrum” we lack of another kind of mathematical language telling people an opposite situation to “zero”------“ something, being,…”.
We need a new number symbol with opposite meaning to zero locating at the opposite side of zero in the “number spectrum” to make up the structural incompleteness of “number spectrum” and to complete the existence of “zero”.
Sincerely yours,
Geng
I have put the following answer in another thread one year ago, but I repeat it with pleasure:
Zermello Fraenkel (ZF) and Zermello Fraenkel with Choice (ZFC) are only proposed Axiom systems. If they prove unconsistent, that does not mean that there are no sets, that there is no mathematics anymore, etc. This only means that a tentative of first order foundation by a system of schemes of axioms was unconsistent, and that the problem is open again. However, it is very likely that a new system of axioms set theory will most probably have the same problem as ZF: we will prove immediately that it is impossible to prove its consistency. So finally we will maybe adopt a position near to that of Bourbaki, in spite of its so called ignorance, see
https://www.dpmms.cam.ac.uk/~ardm/bourbaki.pdf
To sum up: both if ZF is consistent or not, this question remains a never ending story by ist own and intimate nature. However, inspite of the possible instability of such an axiomatic construction, basic objects of mathematics like N, Q, R, C continue to exist and mathematics continue to be done. An incosistency of a given system of axioms does not touch the object for which it was standing. It touches only the problem of finding the right system of axioms for the given object.
Is the set theory (ZFC) consistent?. Available from: https://www.researchgate.net/post/Is_the_set_theory_ZFC_consistent2 [accessed Mar 29, 2015].
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