Learn Basic Postulates in Geometry with Our Experts

For integers 1<=a<=b<=c, construct a graph G with k(G)=a, lamda(G)=b and delta(G)=c.

Show that every two blocks have at most one vertex in common

I was wondering whether there is a proof of SSS Congruence Theorem (and also whether there is one for SAS and ASA Congruence Theorem). In my textbook, they are treated as a postulate, or one that we just accept as truth without basis. Is the 3 theorems for similar triangles really just postulates, or are there any proof of them? I tried searching online but I couldn't find one.

Theorem: If a is a real number, then $a\cdot <!--\; *\; -->0=0$.

1. $a\cdot <!--\; *\; -->0+0=a\cdot <!--\; *\; -->0$ (additive identity postulate)
2. $a\cdot <!--\; *\; -->0=a\cdot <!--\; *\; -->(0+0)$ (substitution principle)
3. $a\cdot <!--\; *\; -->(0+0)=a\cdot <!--\; *\; -->0+a\cdot <!--\; *\; -->0$ (distributive postulate)
4. $a\cdot <!--\; *\; -->0+0=a\cdot <!--\; *\; -->0+a\cdot <!--\; *\; -->0$ I'm lost here, wanna say its the transitive
5. $0+a\cdot <!--\; *\; -->0=a\cdot <!--\; *\; -->0+a\cdot <!--\; *\; -->0$ (commutative postulate of addition)
6. $0=a\cdot <!--\; *\; -->0$ (cancellation property of addition)
7. $a\cdot <!--\; *\; -->0=0$ (symmetric postulate)

So I'm not sure what to put down for the 4th step. The theorem and proof were given and I had to list the postulates for each step.

Is there a form of non-Euclidean geometry in which perpendicular lines never cross, or cross twice or something?

I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :)

In Peano arithmetic addition is usually defined with the following two postulates:
$(1a):p+0=p$
$(2a):p+S(q)=S(p+q)$
Lets say I put the successor term of the second postulate on the left? Namely:
$(1b):p+0=p$
$(2b):S(p)+q=S(p+q)$
Are these two definitions of addition equivalent?

I read that the following can be proved using Bertrand's postulate (there's always a prime between $n$ and $2n$): $\mathrm{\forall <!--\; \forall \; -->}N\in <!--\; \in \; -->\mathbb{N}$, there exists an even integer $k>0$ for which there are at least $N$ prime pairs $p$, $p+k$.
But I have no idea how to prove it. Any help would be much appreciated.
Many thanks.

So I'm reading about the history of hyperbolic geometry and something like this came up: "two thousand years later, people gave up on trying to derive the fifth postulate from the other 4 and begun studying the consequences of the mathematical structure without the fifth postulate. The result is a coherent theory". My question: were there any possibility of incoherence? I understand some postulates are actually definition of objects, then, clearly, other postulates might be dependent on this other postulate, but it doesn't seem to be the case.

I'm struggling with this:
Define $P_m:N\to <!--\; \to \; -->N$ by $P_m(n)$ = nm and $m^n$ as $(P_m{)}^n(1)$
Now prove $k^{m+n}=k^m{k}^n$, and $k^{mn}=(k^m{)}^n$ and $(k^n)(m^n)=(km{)}^n$ for all $k,m,n\in <!--\; \in \; -->N$
Much appreciated! - I can get the first one using induction but the others are allusive...

Statement For every $n>1$ there is always at least one prime $p$ such that $n<p<2n$.
I am curious to know that if I replace that $2n$ by $2n-<!--\; -\; -->\mathrm{ϵ<!--\; ϵ\; -->}$, ($\mathrm{ϵ<!--\; ϵ\; -->}>0$) then what is the $inf(\mathrm{ϵ<!--\; ϵ\; -->})$ so that the inequality still holds, meaning there is always a prime between $n$ and $2n-<!--\; -\; -->\mathrm{ϵ<!--\; ϵ\; -->}$

In Introduction to Metamathematics, Kleene introduces a formal system where the first three postulates in the group for propositional calculus are:
$$\begin{gathered} 1a.A\to <!--\; \to \; -->(B\to <!--\; \to \; -->A) \\ 1b.(A\to <!--\; \to \; -->B)\to <!--\; \to \; -->((A\to <!--\; \to \; -->(B\to <!--\; \to \; -->C))\to <!--\; \to \; -->(A\to <!--\; \to \; -->C)) \\ 2.\frac{A,A\to <!--\; \to \; -->B}{B} \end{gathered}$$
As far as I understand, $1a$ and 2 are typical for Hilbert-style deductive systems, but $1b$ is not. A more traditional choice, serving pretty much the same purpose (e.g. proving $A\to <!--\; \to \; -->A$ to start with) would have been:
$(A\to <!--\; \to \; -->(B\to <!--\; \to \; -->C))\to <!--\; \to \; -->((A\to <!--\; \to \; -->B)\to <!--\; \to \; -->(A\to <!--\; \to \; -->C))$
What is the rationale for the unique choice made for $1b$ in Introduction to Metamathematics?

The Corresponding Angles Postulate states that, when two parallel lines are cut by a transversal , the resulting corresponding angles are congruent. Is this a direct obvious consequence of Euclid's fifth postulate? I think so, because the fifth postulate is basically saying that through a point there is only one line passing parallel to another; am I right? If not, how can you be sure your geometry is consistent having defined parallel lines as those that never meet?

I am beginning real analysis and got stuck on the first page (Peano Postulates). It reads as follows, at least in my textbook.

Axiom 1.2.1 (Peano Postulates). There exists a set $\mathbb{N}$ with an element $1\in <!--\; \in \; -->\mathbb{N}$ and a function $s:\mathbb{N}\to <!--\; \to \; -->\mathbb{N}$ that satisfies the following three properties.
a. There is no $n\in <!--\; \in \; -->\mathbb{N}$ such that $s(n)=1$.
b. The function s is injective.
c. Let $G\subseteq <!--\; \subseteq \; -->\mathbb{N}$ be a set. Suppose that $1\in <!--\; \in \; -->G$, and that $g\in <!--\; \in \; -->G\Rightarrow <!--\; \Rightarrow \; -->s(g)\in <!--\; \in \; -->G$. Then $G=\mathbb{N}$.

Definition 1.2.2. The set of natural numbers, denoted $\mathbb{N}$, is the set the existence of which is given in the Peano Postulates.

My question is: From my understanding of the postulates, we could construct an infinite set which satisfies the three properties. For example, the odd numbers $\{1,3,5,7,\dots <!--\; \dots \; -->\}$, or the powers of 5 $\{1,5,25,625\dots <!--\; \dots \; -->\}$, could be constructed (with a different $s(n)$, of course, since $s(n)$ is not defined in the postulates anyway). How do these properties uniquely identify the set of the natural numbers?

Recall that Bertrand's postulate states that for $n\ge <!--\; \ge \; -->2$ there always exists a prime between $n$ and $2n$. Bertrand's postulate was proved by Chebyshev. Recall also that the harmonic series
$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots <!--\; \cdots \; -->$
and the sum of the reciprocals of the primes
$\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots <!--\; \cdots \; -->$
are divergent, while the sum
$\underset{n=0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}\frac{1}{n^p}$
is convergent for all $p>1$. This would lead one to conjecture something like:

For all $\mathrm{ϵ<!--\; ϵ\; -->}>0$, there exists an $N$ such that if $n>N$, then there exists a prime between $n$ and $(1+\mathrm{ϵ<!--\; ϵ\; -->})n$.

Question: Is this conjecture true? If it is true, is there an expression for $N$ as a function of $\mathrm{ϵ<!--\; ϵ\; -->}$?

By Bertrand's postulate, we know that there exists at least one prime number between $n$ and $2n$ for any $n>1$. In other words, we have
$\mathrm{\pi <!--\; \pi \; -->}(2n)-<!--\; -\; -->\mathrm{\pi <!--\; \pi \; -->}(n)\ge <!--\; \ge \; -->1,$
for any $n>1$. The assertion we would like to prove is that the number of primes between $n$ and $2n$ tends to $\mathrm{\infty <!--\; \infty \; -->}$, if $n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$, that is,
$\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}\mathrm{\pi <!--\; \pi \; -->}(2n)-<!--\; -\; -->\mathrm{\pi <!--\; \pi \; -->}(n)=\mathrm{\infty <!--\; \infty \; -->}.$
Do you see an elegant proof?

Lorenz, Every single line through a point within an angle will meet at least one side of the angle.
I know I have to Show that the parallel postulate 5 implies lorenz, and then lorenz implies parallel postulate 5.
Assume postulate 5 . So we are given AB and a point C not on AB. Choose B on AB draw CD to construct angle ECD= angle BDC.
I just don't get what Lorenz postulate means. Thats where I am getting stuck.

Here are the five postulates:
1. Each pair of points can be joined by one and only one straight line segment.
2. Any straight line segment can be indefinitely extended in either direction.
3. There is exactly one circle of any given radius with any given center.
4. All right angles are congruent to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than two right angles.

Questions:
1. These to me sounds more like something that shouldn't require proving... does it?
2. Why is it important to stress things that are obvious? For example, what other answers can you get when extending a line segment other than it can be extended indefinitely in either direction?
3. Similarly, what space can allow two circle of the same radius and center to be not the same?
4. Saying all right angles are congruent ... isn't that the same as saying all 64.506 degree angles are congruent? Isn't it ANY angle are congruent if they are the same degrees measures from the same reference point (say x-axis)?
5. Why do we need the 5th postulate?

I'm trying to get an overarching understanding of the components of mathematical systems so that in my self study of each category of math I can break them down by their unique aspects, i.e. the operators they use, the major concepts they deal with (i.e. how calculus is about "change"), etc.

As far as my experience with formal math terminology goes, im rather weak, and i get utterly confused by the technicality required in formal definitions.

As a good starting point, I'd like to better understand what the difference is between an axiom, a theorem and a postulate. At my current level of knowledge i would use them interchangeably (lol), however I'm sure one is founded upon the others.

If someone could explain the logical hierarchy/relation between these three it would be greatly appreciated.

I have been recently studying a C.G. Hempel's article on mathematical truth and pointed out his following quotation: "Every concept of mathematics can be defined by means of Peano's three primitives,and every proposition of mathematics can be deduced from the five postulates enriched by the definitions of the non-primitive terms".

I was wondering if it is possible for someone to make a scheme illustrating the sequence of the derivation of the whole theory of mathematics being derived by these postulates.Possibly starting from natural numbers?(notice that Hempel excludes geometry)