equiconsistent to ZFC

Stream: maths

[Huỳnh Trần Khanh]

is it true that to show that a formal system is equiconsistent to ZFC, I have to construct a model of that formal system in ZFC? and is that enough? also do you say "equiconsistent to" or "equiconsistent with" lol

[Huỳnh Trần Khanh]

let's say i have a smart friend that doesn't believe in the consistency of Euclidean geometry. to convince them of its consistency, I have to construct a model of Euclidean geometry by:

• constructing natural numbers
• integers
• rationals
• reals
• analytic geometry

in ZFC and show that the analytic geometry model that I constructed satisfies the axioms of Euclidean geometry
is that correct

[Eric Wieser]

@Huỳnh Trần Khanh, I recommend titling your topics something that summarizes its content rather than something devoid of information like you have here (was: "is my thinking correct"); it increases the chance of someone interested in answering seeing your post, and of you ever finding it again.

You can edit the topic of your above message.

[Huỳnh Trần Khanh]

ah maybe to construct analytic geometry I have to construct calculus and complex numbers too... but you get the idea...

[Kevin Buzzard]

It is almost never true in mathematics that to prove X you have to use method Y.

[Kevin Buzzard]

Your construction of a model of Euclid's axioms is within ZFC, so as long as your friend believes in the consistency of ZFC this looks like a fine proof. In other words, it's a proof of Con(ZFC) -> Con(Euclid)

[Mario Carneiro]

is it true that to show that a formal system is equiconsistent to ZFC, I have to construct a model of that formal system in ZFC? and is that enough?

Actually, if you construct a model of formal system T in ZFC, then you have proved that T is not equiconsistent with ZFC, because you have proven ZFC |- Con(T), or Con(T) < Con(ZFC) (i.e. the consistency strength of T is strictly less than that of ZFC)

[Mario Carneiro]

Equiconsistency of two formal systems means Con(T1) <-> Con(T2)

[Mario Carneiro]

Huỳnh Trần Khanh said:

let's say i have a smart friend that doesn't believe in the consistency of Euclidean geometry. to convince them of its consistency, I have to construct a model of Euclidean geometry by:

• constructing natural numbers
• integers
• rationals
• reals
• analytic geometry

in ZFC and show that the analytic geometry model that I constructed satisfies the axioms of Euclidean geometry
is that correct

This will prove that ZFC |- Con(EuclidGeom), or Con(ZFC) -> Con(EuclidGeom). That is to say, this is a relative consistency proof, of euclidean geometry relative to ZFC. Pretty much all consistency proofs are actually relative consistency proofs, but it means that you can convince your hypothetical friend not that euclidean geometry is consistent, but that if they doubt the consistency of euclidean geometry then they must also doubt the consistency of ZFC. (Usually this is still useful, since ZFC will generally be more studied than the theory being proven consistent, in this case euclidean geometry.)

[Huỳnh Trần Khanh]

lol thanks and I'll say "equiconsistent with" from now on because everyone says equiconsistent with and I'm the odd one out :big_smile: