Since euclidean geometry is a formal system,...

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Since euclidean geometry is a formal system, and formal systems are either complete or incomplete:

https://en.wikipedia.org/wiki/Completeness_(logic)

Is euclidean geometry complete?

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>>7772575

No.

http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Greenberg2011.pdf

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>>7772726

Does it give any examples? Or does it simply prove that such statements must exist?

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>>7772766

that peper the anon posted

http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Greenberg2011.pdf

discusses also formalizations that are complete - of course they are slightly different theories then

https://en.wikipedia.org/wiki/Hilbert's_axioms

https://en.wikipedia.org/wiki/Tarski's_axioms

https://en.wikipedia.org/wiki/Birkhoff's_axioms

..

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>>7772787

Wait, so Birkhoff's formalization is complete? Is it just Euclid's set of axioms that lead to an incomplete plane geometry?

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>>7772575

I'm pretty sure it's incomplete, since it seems like you should be able to do something like Godel-numbering.

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>>7772726

This is constructible geometry tho. The axiomatic system describing it is compass and straightedge geometry. This is probably not what OP was referring to.

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>>7772790

I'm not that guy but I think the problem is that there's some confusion here between the axiomatic system and models of it.

In particular, the euclidean geometry that we all picture in our heads with lines and points is really just a model of these systems (all of them). This means it's an example of a thing where all the axioms hold. Unfortunately, whenever you're talking about a model it's possible to make statements about the model that might not be provable of the system itself. This just means that there might exist other models where the statement is not true.

Suppose I create a formal system for describing rectangles. Now suppose that in my head when I picture a rectangle I'm actually picturing a square. Sure, my square is an example of a rectangle, and everything I can prove about rectangles will also be true about squares. However the opposite is not the case. There are things that are true about squares that are not provable in the rectangle system (like all sides are of equal length). Worse still, I could assume that these things are so obviously true that I might never bother to show the work and might never realize that I'm missing a much bigger picture.

This is what happened with constructive geometry (compass and straightedge). It is also what happened with Euclid's axioms (Google Pasch's theorem). I don't think it's currently known if the other systems have this problem as well.

When a system does not have this problem it is because all models of the system are isomorphic. We call such systems categorical. Some examples of such systems include the peano axioms (though I think some doubt was raised about this recently due to a paper I read) and the real number axiom system.

Don't ask if the system is complete. Ask if you truly understand it and grasp the big picture.

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>>7772575

I heard Taylor Swift was only into circumcised penises. If true, makes her somewhat unattractive.

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>>7772882

>Unfortunately, whenever you're talking about a model it's possible to make statements about the model that might not be provable of the system itself.

does this mean that not all proofs can be converted back to pure syntax ? why do proofs on the semantic side hold only in the models in which they hold and not in other models ?

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>>7773056

That was the case with Pasch's axiom. However no one noticed for about two thousand years because the statement was so simple and obviously true that no one ever bothered to try to write it in pure syntax. I imagine that anyone who even questioned it probably wouldn't been ridiculed and called retarded for not seeing the obvious.

Right now people are somewhat more careful, but there's still a lot of informal mathematicians who make "obvious" statements. I do not like this but it is the norm.

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>>7773072

but I do not understand the difference between a theory and a model. the model has everything that the theory has but has also more ?? what is this more ??

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>>7773081

A theory (often called axiomatic system) is just a set of sentences (often called axioms) in your formal system (formal language + logic, proof system).

This probably seems confusing. So let's look at my favorite example, incidence geometry. We have three axioms:

Axiom 1) For every pair of distinct points [math]P[/math] and [math]Q[/math] there exists exactly one line [math]l[/math] such that both [math]P[/math] and [math]Q[/math] lie on it.

Axiom 2) For every line [math]l[/math] there exist at least two distinct points [math]P[/math] and [math]Q[/math] such that both [math]P[/math] and [math]Q[/math] lie on [math]l[/math].

Axiom 3) There exist three points that do not all lie on any one line.

Now, these axioms aren't written in a formal language, but we could easily do that if we wanted to (we would just write unary predicates for "is a point" and "is a line" as well as a binary predicate for "is incident with").

Note that we have to be fairly careful when dealing with things abstractly like this. For instance we never said that all objects are lines or points, nor did we say that an object can't be both a line and a point at the same time. Either way, these three axioms give us the theory of incidence geometry. We can prove many things about it and we can think up many examples of it. We can even give definitions like:

Definition 1) Three points [math]A[/math], [math]B[/math], and [math]C[/math] are [math]colinear[/math] if there exists one line [math]l[/math] such that all three of the points [math]A[/math], [math]B[/math], and [math]C[/math] all lie on [math]l[/math].

Definition 2) The points are [math]noncolinear[/math] if there is no such line [math]l[/math].

(cont.)

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>>7773141

(cont.)

Now lets begin talking about models of this theory. We can interpret our points as cities, our lines as railroads, and then say that points lie on a line if there is a train station in that city for that railroad. Now, for such a model to satisfy our theory it must satisfy our axioms, so this likely wouldn't work for just any set of cities and railroads but we can come up with an example where it can. Let's say our cities are named A, B, and C. Then lets say we have three railroads. The first has stations at A and B, the second has stations at B and C, and the third has stations at A and C. This model satisfies our theory. Note, that we could then say some extra things about this model like "there exist only three points". It's true with our cities but not necessarily true of incidence geometry.

If this is too informal or unsatisfactory for you then we could construct our model inside set theory as well. Just let A, B, and C, be sets, then lines are subsets of [math]\{A, B, C\}[/math] of cardinality two. Then a point is incident with a line if that point is an element of the line. This is typically called the three-point geometry. There exists a four-point geometry, five-point geometry, and so on.

We can describe many more models as well.

The Fano plane is another model of incidence geometry.

https://en.wikipedia.org/wiki/Fano_plane

The Cartesian plane [math]\mathbb{R}^2[/math] is a model of incidence geometry (our [math]points[/math] are points in [math]\mathbb{R}^2[/math] and our [math]lines[/math] are sets of points [math](x,y)[/math] satisfying an equation [math]ax + by + c = 0[/math] where [math]a,b\neq 0[/math] and [math]a,b,c\in \mathbb{R}[/math].

We can consider a sphere and construct a geometry on it's surface in a similar way. [math]Points[/math] are points on the surface and [math]lines[/math] are sets of points along the surface that effectively cut the sphere in half (these are also typically formalized with equations).

(cont.)

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>>7773145

The Klein disk model is a model of incidence geometry.

https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model#The_Klein_disk_model

Now for something more interesting. First we need a definition.

Definition 3) Two lines [math]l[/math] and [math]m[/math] are said to be parallel if there is no point [math]P[/math] such that [math]P[/math] lies on both [math]l[/math] and [math]m[/math].

Consider these three parallel postulates corresponding to Euclidean geometry, Elliptic geometry, and Hyperbolic geometry.

Euclidean Postulate) For every line [math]l[/math] and for every point [math]P[/math] that does not lie on [math]l[/math], there is exactly one line [math]m[/math] such that [math]P[/math] lies on [math]m[/math] and [math]m[/math] is parallel to [math]l[/math].

Elliptic Postulate) For every line [math]l[/math] and for every point [math]P[/math] that does not lie on [math]l[/math], there is no line [math]m[/math] such that [math]P[/math] lies on [math]m[/math] and [math]m[/math] is parallel to [math]l[/math].

Hyperbolic Postulate) For every line [math]l[/math] and for every point [math]P[/math] that does not lie on [math]l[/math], there are at least two lines [math]m[/math] and [math]n[/math] such that [math]P[/math] lies on both [math]m[/math] and [math]n[/math] and both [math]m[/math] and [math]n[/math] are parallel to [math]l[/math].

These postulates are not provable in incidence geometry because there exist models satisfying each one but not satisfying the others. In this case we say that the postulates are "independent" of Incidence geometry (just like how the axiom of choice and the continuum hypothesis are independent of ZF). This does not mean that they are false, nor does it mean that they are true. Just that there are cases where they are false and cases where they are true. To truly understand we must look at the bigger picture.

With regards to Euclids axioms for geometry. Take a look at

https://en.wikipedia.org/wiki/Pasch%27s_axiom

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>>7773145

>>The Cartesian plane R2\mathbb{R}^2 is a model of incidence geometry (our pointspoints are points in R2\mathbb{R}^2 and our lineslines are sets of points (x,y)(x,y) satisfying an equation ax+by+c=0ax + by + c = 0 where a,b≠0a,b\neq 0 and a,b,c∈Ra,b,c\in \mathbb{R}.

this one makes me understand that actually, given a logic, given a theory in that logic, a model for this theory in this logic can be informally an extension of another model [of another theory, in the same logic]. here you already have a model of the real line, stemming from some theory ''of the real line'' in some logic. then you like this model and you see if your new theory of incidence geometry can be interpreted by the model of the real line, if the theory of incidence geometry can be part of the model of the real line.

is this correct. ?

the observation is then that your model with the three cities A, B, C is clearly more minimal than whatever machinery coming already with the model of the real line, in which you want to express your theory of incidence geometry.

the three cities appears as the most minimal model for the theory, because in this model, there is nothing more nor less axioms than in the theory itself.

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>>7772892

>having an uncut, uncivilized, barbaric reproductive organ

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>>7772882

>Some examples of such systems include the peano axioms

Second order Peano axioms. The first order formulation is not categorical.

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>>7773228

>having a dismembered, mutilated, permanently destroyed reproductive organ

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>>7773424

Uncut penises are more prone to catching diseases and fungi and disfunctional issues due to the lack of hygene

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>>7773435

Only if you shower like once a week. Stop being a disgusting slob.

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>>7773446

it's a comparison with equal shower intervals obviously retard. the uncut skin allows bacteria to cultivate and grow, like the lardflaps of a fat guys belly. Theres a reason why people cut it.

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>>7773450

>Probably religious reasons. Shower like a normal person and that bacterial growth won't be a problem.

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>>7773480

So what? It makes no practical difference if you shower like a normal person.

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>>7772892

Some people have standards or simply don't have foreskin fetish. Deal with it.

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>>7773228

I'd say it's fairly barbaric and uncivilized to strap newborn infants down and to cut off pieces of their genitals without their consent.

>>7773484

>Some people don't have a natural anatomy fetish.

People are into what they're raised around. That is unfortunately the extent of their faculty for thought.

I bet if she gave some foreskin a try, her thinking would change pretty quick ;)

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>>7773450

That's not how it works, anon.

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>>7773480

>Even if you do wash the same time as a cut person, the area under the skin is relatively more prone to grow bacteria and fungi compared to a cut one.

men who rigged the stats are precisely those who are dirty

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>>7773480

The inner foreskin is a mucous membrane, and has its own immune system-like functions. And I can assure you, anaerobic fungi are the last things you would find under there.

Your own penis, cut or not, has bacteria on it. Every female you might or might not have ever fucked had bacteria in there, and under her clitoral hood. All of your skin is covered in micro-organisms and bacteria. And that's fine, showering and washing too much is doing you no favors whatsoever.

I hope the US can be fixed someday. People's idea of health and hygiene is just plain delusion.

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In light of the nice explanation of the anon above (for where it comes from that a theory has sentences that can't be proven trur or false ... Because there can be both, models that satisfy it and models where its false) it's worth pointing out how Gödel result on the theories of numbers is a much stronger hammer: he proved that not only is the theory of Peano arithmrtic incomplete (assuming its consistent), but he also showed it's not completable! He constructs an unprovable statement w.r.t. the current, and if you try to fix the theory by more axioms, then another sentence will be the new unprovable one.

Regarding the question above (already answered), second oder Peano arithmetic is the categorical one.

Actually, Gödels completeness theorem (the the other big theorem) for first order logic says all statements true in all model are provable.

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https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem

The positive result (and one of the reasons first order logic is prefered over second oder logics, even thuogh the former needs sets (or something similr) to raise expressiveness)

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Euclid's geometry is not a formal system in the modern sense so the question of completeness or decidability is not answerable in the strict sense. Hilbert's system of axioms is the first modern formal system of geometry, but his system was second order and therefore could encode enough arithmetic to be incomplete/undecidable. Tarski's system is first order and is compete/decidable. So the answer is yes and no and maybe; depends on your system.

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>>7773956

>second oder Peano arithmetic

what does it look like compared to the first order one ?

I do not follow what differs between first order and second order logic [in terms of sets, once you look at models of sets]

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>>7774056

I don't think I can say it more straight forewardly than the wiki articles on first-order logic and first-otder arithmetic.

The tl;dr is that second order logic allows for forall and exists binding subsets of N, not just numbers in N. The syntax is thus more expressive (almost as expressive as if you do first order set theory with all its overblown nonconstructuve axioms)

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>>7774056

I'm not sure, but I think that second order logic basically allows you to quantify predicates. In other words, you can write sentences such as "there exists a property that satisfies..." which is impossible within first order logic.

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>>7772892

>You will never show T Swift the world of pleasures afforded by foreskin

>You will never debate T Swift in a cut vs uncut thread

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>>7776055

>why even live?

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