All Questions
Tagged with peano-axioms definition
22
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3
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How to find what a definition defines?
Are these definitions of the "+" and "mod" operators?
$m+0=m$........(1)
$0 \;\text{mod} \;2 = 0$.....(2)
To me, (1) defines the identity property of zero and (2) defines zero as ...
- 437
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3
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Operation 'Referencing' In Abstract Algebra
I recently started an Abstract Algebra course so I may not know all of the 'necessary' vocabulary to fully express my question, but here is my attempt. First I need to provide some background thinking....
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Peano' s Systems
Lets consider $(A,g,a_{0})$ , where $A$ is a set, $g$ is a function and $a_{0}$ is the first element of $A$. If $(A,g,a_{0})$ follows all the Peano's axioms, than we can find an isomorphism f between $...
- 153
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Building product in $\Bbb N$ using the function $s: n\mapsto n+1$
using the Peano's axioms we can give a description of the set of natural numbers.
Let's consider the functions
$s: \Bbb N\to \Bbb N$ defined by $s(n)=n+1$
and
$f^n=\begin{cases}
id, &\text{ if } n=...
- 153
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2
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45
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Existence of closures of sets
I am reading Stillwell's Elements of Algebra. And in Chapter 1, he introduces the real quadratic closure of $\mathbb Q$ as
the set of the numbers obtainable form $\mathbb Q$ by square roots of ...
- 4,119
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How to define a large prime number (for example, $7919$)?
The set of natural numbers may be defined using Peano Axioms:
Under this definition of natural numbers, one may define $1$ as $0^+$, $2$ as $(0^+)^+$, $7$ as $((((((0^+)^+)^+)^+)^+)^+)^+$, etc.
My ...
- 5,451
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1
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Really confused about the relationship between set theory, functions, ZFC, Peano axioms, etc.
I don't understand how everything relates. It seems that ZFC is a "first order theory" with axioms described in the language of first order logic, and it can recreate all the same axioms of Peano ...
- 175
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Why isn't addition defined by $a+0=a$ and $a+succ(b)=succ(a)+b$?
The rule for addition is stated in the Peano axioms as:
$a + 0 = a$
$a + succ(b) = succ(a + b)$
But why couldn't we define it as
$a + 0 = a$
$a + succ(b) = succ(a) + b$
for example $3 + 4 = 4 + ...
- 103
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I am confused about how we should state the Peano axioms
I am really confused if the Peano axioms are supposed to be strictly a set theory / first order thing or how are we supposed to state them. In English? Can we use purely logical expressions? What ...
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Why does induction only allow numbers connected to $0$ to be natural?
When learning Peano axioms normally we use induction to suggest that if a property $P(0)$ is true, and if $P(k)$ being true implies $P(k + 1)$ being true (for natural number $k$), then if $P$ is the ...
2
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In Peano's Axioms are the uniqueness of the successor and $x^{\prime}=y^{\prime}\implies{x=y}$ redundant?
In Peano's Axioms are the uniqueness of the successor and the property $x^{\prime}=y^{\prime}\implies{x=y}$ redundant?
This seems obvious to me, but I may be missing something. In the various forms ...
- 3,349
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Difference between first and second order induction?
Can anyone explain the difference between induction as it's stated in first order logic and that from second order logic? I don't understand the difference as it pertains to things like Peano axioms.
- 5,651
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Why do we need the following set to have an infinite set?
By Dedekind's definition:
Definition: We say that some set $S$ is infinite if there exists an injection $f:S\rightarrow S$ such that $\operatorname{Im}(f)\neq S$ (or $f(S)\neq S)$.
Now, the book ...
- 2,885
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Order of the natural numbers
The set of natural numbers as given from the Peano axioms $(N,S)$ has an order.
I saw in wikipedia that $(N,+)$ is a commutative monoid, but since the naturals have an order structure by ...
- 313
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"Recursive definitions" in Tao's Analysis Vol I
I am totally confused when Tao gets into recursive definitions (page 26).
Paraphrasing, the axioms of natural numbers let us define sequences recursively. Suppose we want to build a sequence $a_0, ...
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