Help me understand: "is it consistent with itself?"

  • #1

Main Question or Discussion Point

Hello.
The following is the content taken from here. It is a summary or map of mathematics.

"Foundation:
This area tries to work out with properties of mathematics itself and asks what the basis of all rules of mathematics is and is there complete set of fundamental rules or axioms from which all mathematics comes from and can we prove it is consistent with itself? Mathematical theory, set theory and category theory try to answer this and the famous result in mathematical logic is Gödel incompleteness theorem which for most people means that mathematics does not have complete and consistent set of axioms which means it is all kind of made up by humans."

My question is:
Why do I need to talk about consistency in mathematics? What does it mean if mathematics is said to have consistency with itself? I am unable to understand what the text highlighted in red (above) means.

Thank you.
 
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Answers and Replies

  • #2
FactChecker
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There would be a logic problem if the basic assumptions of mathematics contradicted each other. That must be avoided if we want to trust the logic.
 
  • #3
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My question is:
Why do I need to talk about consistency in mathematics? What does it mean if mathematics is said to have consistency with itself? I am unable to understand what the text highlighted in red (above) means.
It means, that given a certain system of axioms (assumed facts and given deduction rules), you cannot deduce a contradiction.

This is where the crisis in set theory at the beginning of the 20th century came from: What is the set ##\mathbb{S}## of all sets, that doesn't contain itself as an element? Does ##\mathbb{S}## belong to ##\mathbb{S}## or not? This led to a contradiction in the definition of a set. But as soon as you have a statement in your system, which is simultaneously true and false, i.e. a contradiction, you can derive literally everything from it, including all false statements. Thus it would be a worthless system. As a consequence you are forced to find a system, in which such contradictions cannot occur, which is consistent with itself. In the example of set theory, the short answer is the introduction of the concept of classes. (@all: I know that this is a very short version of what actually happened, see the link above.)
 
  • #4
It means, that given a certain system of axioms (assumed facts and given deduction rules), you cannot deduce a contradiction.
So, to avoid contradictions, one has to use a correct system of axioms?
 
  • #5
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So, to avoid contradictions, one has to use a correct system of axioms?
Ideally, yes. In mathematical reality this is not always the case. E.g. David Hilbert's second problem was the question "Are the arithmetic axioms without contradictions?" (1900) and Kurt Gödel could prove (1931) that this cannot be decided by arithmetic axioms alone. So there are examples where axioms work quite well every single day, but we cannot show their consistency without the application of meta-mathematical principles.

There are directions in mathematics, that try to deal with only constructive methods in order to avoid those troubles.

Another interesting aspect in mathematics is the axiom of choice. Kurt Gödel has proven (1938) that the axiom of choice doesn't lead to a contradiction to the other axioms of set theory by Zermelo-Fraenkel. Paul Cohen could show (1963) that the negation of the axiom of choice won't lead to contradictions either. Therefore you could assume the axiom of choice to be true or to be false without affecting the other axioms.

All this shows, that it isn't actually a trivial task to prove consistency for complex axiomatic systems and "to use a correct system of axioms" might lead to heavy debates about what this should mean. The opposite is easier by nature, as to prove inconsistency, one only needs a single contradiction.
 
  • #6
hilbert2
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So, to avoid contradictions, one has to use a correct system of axioms?
As an example, let's say that I attempt to make a system of real numbers that has all the usual rules like

##x + y = y + x##,
##xy=yx##

and so on, but I add another rule that there are two numbers, ##0## and ##0'##, with the zero property

##x+0=x## for any ##x##.

Now I can deduce a contradiction by writing

##0 = 0 + 0' = 0' + 0 = 0'##,

which tells that the zeroes would actually be the same number.

Real numbers with two zeroes would be an obvious contradictory system, but you can also imagine a system that looks ok, but can be proved to be contradictory with some very difficult reasoning.
 
  • #7
There would be a logic problem if the basic assumptions of mathematics contradicted each other. That must be avoided if we want to trust the logic.
Real numbers with two zeroes would be an obvious contradictory system, but you can also imagine a system that looks ok, but can be proved to be contradictory with some very difficult reasoning.
Assume that there is a perfect contradictory system then what does mathematics become?
 
  • #8
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Assume that there is a perfect contradictory system then what does mathematics become?
But as soon as you have a statement in your system, which is simultaneously true and false, i.e. a contradiction, you can derive literally everything from it, including all false statements.
 
  • #9
Stephen Tashi
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So, to avoid contradictions, one has to use a correct system of axioms?
What shall we mean by a "correct" system of axioms?

If we are thinking about applying mathematics to the physical world then the axioms that describe one physical phenomena may be incorrect as a description for a different phenomena. So it isn't useful to view the definition of correct axioms" as meaning "true" or "real". If we are thinking about a mathematical system without regard to whether it applies to any particular physical situation then one criteria for "correct axioms" is merely that they are not self-contradictory.
 
  • #10
What shall we mean by a "correct" system of axioms?
By saying "correct system of axioms" I mean statements that are philosophically correct.

If we are thinking about a mathematical system without regard to whether it applies to any particular physical situation then one criteria for "correct axioms" is merely that they are not self-contradictory.
Not just not self-contradictory but also the statements that switch between correct and wrong by reasoning or by having rule that does that. A correct statement cannot remain correct if it is not philosophically justified and a wrong statement cannot remain wrong if it not philosophically justified.
 
  • #11
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I think that terms like "philosophically justified" and "philosophically correct" are hard to define or defend. Analytical philosophers like Bertrand Russel would insist on strict definitions. Anything that goes outside of formal logic and math is treacherous and highly debatable.
 
  • #12
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Anything that goes outside of formal logic and math is treacherous and highly debatable.
I think that a meta-level can be properly defined to allow to speak about an axiomatic system. This led me to the following question: Is the statement, an axiomatic system isn't self-contradictory, already a statement on the first meta-level?
 
  • #13
hilbert2
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This made me get an idea of a self referential axiom system that contains rules how it would have to be changed if a contradiction is found... Looks like something like that has been thought of before: ftp://ftp.idsia.ch/pub/juergen/gmAGI.pdf .
 
  • #14
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This made me get an idea of a self referential axiom system that contains rules how it would have to be changed if a contradiction is found... Looks like something like that has been thought of before: ftp://ftp.idsia.ch/pub/juergen/gmAGI.pdf .
Could be a reasonable rule to design AIs.
 
  • #15
QuantumQuest
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Just to add to all the above great posts, John von Neumann did great work on set theory, influenced by Georg Cantor., Ernst Zermelo and the critiques of Zermelo's set theory given by Fraenkel and Skolem. Von Neumann worked on the deficiencies of Zermelo's set theory introducing some ideas to remedy them. The result was NBG (von Neumann–Bernays–Gödel) set theory which was a canonical extension to Zermelo–Fraenkel set theory (ZFC). NBG unlike ZFC can be finitely axiomatized. See Wikipedia article for more.
 
  • #16
I think that terms like "philosophically justified" and "philosophically correct" are hard to define or defend.
I think, we are far only by a rule between two statements to achieve the conclusion of the debate.
 

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