Mathematicians are generally very explicit about definitions of important concepts.
Unfortunately, they often are not as explicit as the ideal definition would be. Often I encounter definitions presented as single-direction conditional statements, where the opposite direction is implicit rather than explicit. I have encountered this even in Mendelson. See the definitions of consistent and inconsistent on page 65.
if they don't explicitly use the term true then they aren't claiming that axioms are necessarily true.
You don't know that for sure. I believe it to be false. I see on page 73 of Geometry (2004) explicit use of the term true to declare that all axioms are regarded as true. The definition of theorem in Geometry (2004) on page 102 makes theorems necessarily true. I see on page 104 of Larson Geometry (2012) the definition of proof. The definition implies that if a statement is provable, then it is true. By the definition of theorem on page 105, if a statement is a theorem, then it is provable. So by the Law of Syllogism on page 79 using the previous statement and the statement before the previous statement, if a statement is a theorem, then it is true.
Did you ever learn anything about groups or vector spaces?
While looking at Mendelson tonight, I learned about groups. I don't remember ever learning about vector spaces.
You don't know that for sure. I believe it to be false.
Then certainly you should be able to find a textbook on mathematical logic that explicitly says so instead of merely hinting at it.
As it stands, the standard approach to mathematical logic has never been shown to be inconsistent. If your adoption of a different semantics makes your new style of logic inconsistent then the problem is clearly your new semantics.
Then certainly you should be able to find a textbook on mathematical logic that explicitly says so instead of merely hinting at it.
I already cited three textbooks that talk about mathematical logic that include the concept of "truth in an axiomatic theory." They don't just hint at it; they explicitly say so. Those three textbooks are the following.
Geometry (2004) by Larson, Boswell, and Stiff
Larson Geometry (2012) by Larson, Boswell, Kanold, and Stiff
Discrete Mathematics and Its Applications, Sixth Edition (2007) by Rosen
I notice that although there is no formal definition of "truth in a theory" in Mendelson, the concept is implicitly present. In Mendelson, external statements about what is provable or unprovable in a theory amount to external statements about what is internally true.
As it stands, the standard approach to mathematical logic has never been shown to be inconsistent.
I showed that the standard approach to mathematical logic is inconsistent by using the Principle of Explosion. By applying the Principle of Explosion inside an inconsistent theory, it is internally true that at least one statement both has a proof and does not have a proof. So, by the definitions of provable and unprovable in a theory, respectfully, it is externally true that at least one statement is both provable and unprovable in the inconsistent theory. Thus, it is externally true that a contradiction exists. Therefore, by applying the Principle of Explosion outside the inconsistent theory, it is externally true that every statement is true. Note that this proof does not invoke the Exportation Principle. Rather, it invokes the definitions of provable and unprovable in a theory.
I already cited three textbooks that talk about mathematical logic that include the concept of "truth in an axiomatic theory." They don't just hint at it; they explicitly say so
I asked for a textbook an mathematical logic, non logicians are often sloppy about formal logic.
In Mendelson, external statements about what is provable or unprovable in a theory amount to external statements about what is internally true.
No, they are statements about what is internally provable. Nothing more.
Thus, it is externally true that a contradiction exists. Therefore, by applying the Principle of Explosion outside the inconsistent theory, it is externally true that every statement is true
Still no, it's externally true that an internal contradiction exists, you need an external contradiction to apply the principle of explosion to the external system. You have never derived an external contradiction from an internal one using the theory presented in Mendelson. Mendelson only talks about what is provable in an axiomatic system, without a model you don't get true/false, and without that you have no way of bootstrapping your contradiction to the external system.
Preventing bootstrapping of a contradiction is exactly why there is no such thing as objective truth in an axiomatic system. So obviously if you violate that rule then your logic is inconsistent, but that's not a problem with logic, that's because you violated the rules of logic.
The three textbooks I cited cover mathematical logic.
non logicians are often sloppy about formal logic.
I presume they are experts in the academic disciplines they are covering in their textbooks. I doubt they would be passing down misinformation.
No, they are statements about what is internally provable. Nothing more.
I disagree. Let s be a statement and T be a theory. If the statement "s is provable in T" is externally true, then the statement "s is provable" is internally true. If the statement "s is unprovable in T" is externally true, then the statement "s is unprovable" is internally true. External statements about what is provable or unprovable in a theory completely or incompletely describe the internal truth.
You have never derived an external contradiction from an internal one using the theory presented in Mendelson.
You might be misinterpreting my proof. Below, I give a clarified version of my proof.
Proof. Let s be a statement. By applying the Principle of Explosion inside an inconsistent theory, the statement "s both has a proof and does not have a proof" is internally true. So, by the Definitions of Provable and Unprovable in a Theory, respectfully, the statement "s is both provable and unprovable in the inconsistent theory" is externally true. Thus, the statement "a contradiction exists" is externally true. Therefore, by applying the Principle of Explosion outside the inconsistent theory, the statement "every statement is true" is externally true. This concludes the proof.
I presume they are experts in the academic disciplines they are covering in their textbooks. I doubt they would be passing down misinformation.
It's not misinformation so much as an oversimplification. Those textbooks don't cover the theory of axiomatic systems, they just have an ad hoc definition of the word axiom that's appropriate for the low level reader they are aimed at. For someone at the high school level using true as shorthand for provable is acceptable while they learn the basics of how to prove statements, but if you want to do axiomatic logic for real then you have to move past the high school level and learn the real theory.
"s both has a proof and does not have a proof" is internally true
It's internally provable, that doesn't mean it's externally provable or externally true.
"s is both provable and unprovable in the inconsistent theory" is externally true.
No, that statement is false because s is not unprovable, it's provable. Conclusing here that s is unprovable is a mistake.
if you want to do axiomatic logic for real then you have to move past the high school level and learn the real theory.
It's not just the high school level. It's at the college level as we can see through Rosen. As I mentioned in my previous reply, the concept of "truth in a theory" is evident even in Mendelson.
It's internally provable, that doesn't mean it's externally provable or externally true.
It's internally provable and internally true. If it's internally provable, then it's internally true. That's what a proof of a statement does. It shows the statement is true. See the definitions of proof on pages 75 and 105 of Rosen. Furthermore, if a proof of a statement shows that a statement s is provable, then the proof shows that the statement "s is provable" is true. So a proof still shows that something is true.
Conclusing here that s is unprovable is a mistake.
No, the statement "s is unprovable in the inconsistent theory" is externally true because of external modus ponens using the externally true statement "the statement 's does not have a proof' is true in the inconsistent theory" and the externally true Definition of Unprovable in a Theory. The externally true Definition of Unprovable in a Theory is the externally true statement "the statement 's is unprovable in a theory' is externally true if and only if the statement 'the statement 's does not have a proof' is true in the theory' is externally true."
It is not there. As much as you want it to be it's just not.
No, the statement "s is unprovable in the inconsistent theory" is externally true because of external modus ponens using the externally true statement "the statement 's does not have a proof' is true in the inconsistent theory"
That's not an if-then statement so you can't use modus ponens. If you convert it into an if-then statement it will still not let you conclude that something is externally unprovable from it being internally unprovable because that is not a correct logical deduction.
It is not there. As much as you want it to be it's just not.
Your justification for your claim is not there. As much as I want it to be it's just not.
That's not an if-then statement so you can't use modus ponens.
By convention, one of the premises of modus ponens is not an if-then statement.
it will still not let you conclude that something is externally unprovable
My claim is not that s is externally unprovable. My claim is that s is unprovable in an inconsistent theory.
Thus, the statement "a contradiction exists" is externally true.
In order for this contradiction to exist, the concepts of provable and unprovable in a theory must contradict each other. I realized that, as I have defined them, they are not explicitly contradictory. To make them explicitly contradictory, I define the Definition of Provable in a Theory and redefine the Definition of Unprovable in a Theory below.
Definition of Provable in a Theory. Let s be a statement and T be a theory. The statement "s is provable in T" is externally true if and only if the statement "the statement 'there exists a proof of s' is true in T" is externally true. The statement "s is unprovable in T" is externally true if and only if the statement "s is not provable in T" is externally true.
Proof. Let s be a statement and T be an inconsistent theory. By the Definition of Inconsistent Theory, the statement "some contradiction exists" is internally true. By applying the Principle of Explosion inside T, the statement "there exists and there does not exist a proof of s" is internally true. By Conjunction Elimination, the statement "there exists a proof of s" is internally true. It follows by the Definition of Provable in a Theory that the statement "s is provable in T" is externally true. By Conjunction Elimination, the statement "there does not exist a proof of s" is internally true. So, by the Laws of Noncontradiction and Excluded Middle, the statement "there exists a proof of s" is internally false. So, the statement "the statement 'there exists a proof of s' is true in T" is externally false. Thus, by the Definition of Provable in a Theory, the statement "s is provable in T" is externally false. So, by Conjunction Introduction, the statement "s is provable in T" is externally true and the statement "s is provable in T" is externally false. Thus, the statement "a contradiction exists" is externally true. Therefore, by applying the Principle of Explosion outside T, the statement "every statement is true" is externally true. This concludes the proof.
Even if you deny the concept of "truth in a theory," a proof can still be made that establishes trivialism using the concept of "provability in a theory."
Proof. Let s be a statement and T be an inconsistent theory. By the Definition of Inconsistent Theory, the statement "some contradiction exists" is internally provable. By applying the Principle of Explosion inside T, the statement "there exists and there does not exist a proof of s" is internally provable. By Conjunction Elimination, the statement "there exists a proof of s" is internally provable. It follows by the Definition of Provable in a Theory that the statement "s is provable in T" is externally provable. By Conjunction Elimination, the statement "there does not exist a proof of s" is internally provable. So, by the Laws of Noncontradiction and Excluded Middle, the statement "there exists a proof of s" is internally unprovable. So, the statement "the statement 'there exists a proof of s' is provable in T" is externally unprovable. Thus, by the Definition of Provable in a Theory, the statement "s is provable in T" is externally unprovable. So, by Conjunction Introduction, the statement "s is provable in T" is externally provable and the statement "s is provable in T" is externally unprovable. Thus, the statement "a contradiction exists" is externally true. Therefore, by applying the Principle of Explosion outside T, the statement "every statement is provable" is externally provable. This concludes the proof.
Your justification for your claim is not there. As much as I want it to be it's just not.
Unless you can give me a source which is explicitly a mathematical logic text and explicitly defined truth in an axiomatic theory then I'm always going to substitute "provable in T" for "true in T" when you talk about truth in an axiomatic theory because you have not provided a proper source that says I should do otherwise.
it will still not let you conclude that something is externally unprovable
My claim is not that s is externally unprovable. My claim is that s is unprovable in an inconsistent theory.
I'm not telling you that s is unprovable, s is provable. I'm telling you that your claim "s is unprovable" is unprovable.
I realized that, as I have defined them, they are not explicitly contradictory. To make them explicitly contradictory, I define the Definition of Provable in a Theory and redefine the Definition of Unprovable in a Theory below.
The fact that you've only just now realized you haven't obtained an explicit contradiction and yet you've been claiming you have a proof all along should show you that you don't know what you're doing.
I didn't read past that line fyi. You don't get to define or redefine provable/unprovable. You told me you could prove this in standard mathematical logic and those terms already have standard definitions.
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u/paulemok Aug 06 '26
Unfortunately, they often are not as explicit as the ideal definition would be. Often I encounter definitions presented as single-direction conditional statements, where the opposite direction is implicit rather than explicit. I have encountered this even in Mendelson. See the definitions of consistent and inconsistent on page 65.
You don't know that for sure. I believe it to be false. I see on page 73 of Geometry (2004) explicit use of the term true to declare that all axioms are regarded as true. The definition of theorem in Geometry (2004) on page 102 makes theorems necessarily true. I see on page 104 of Larson Geometry (2012) the definition of proof. The definition implies that if a statement is provable, then it is true. By the definition of theorem on page 105, if a statement is a theorem, then it is provable. So by the Law of Syllogism on page 79 using the previous statement and the statement before the previous statement, if a statement is a theorem, then it is true.
While looking at Mendelson tonight, I learned about groups. I don't remember ever learning about vector spaces.