I want to construct exponentiation over $\mathbb{R}$.
$$^\wedge:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$$
Using a strictly constructive approach $\mathbb{R}=\{[(x_n)]|(x_n)\in\mathbb{Q}^\mathbb{N}\}$, aka each real number is an equivalence class of rational Cauchy sequences, this implies $\mathbb{Q}\nsubseteq\mathbb{R}$ but each $\frac{a}{b}\in\mathbb{Q}$ is embedded with $[(x_n)]\in\mathbb{R}$ s.t. $\forall n\in\mathbb{N}(x_n=\frac{a}{b})$.
I follow this post for part of this construction but run into a problem below.
Formal definition of numbers with real exponents
NOTE
Rational exponentiation is only defined for integer exponents $^\wedge:\mathbb{Q}\times\mathbb{Z}\to\mathbb{Q}$, where $\mathbb{Z}$ is the integer embedding in $\mathbb{Q}$.
Therefore for $^\wedge:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$, we don't have the tools of rational exponents, radication, or logarithms.
Likewise, since $^\wedge:\mathbb{Q}\times\mathbb{Z}\to\mathbb{Q}$ isn't defined on the entirety of $\mathbb{Q}^2$, the Cauchy construction of real addition and multiplication wouldn't work for exponents as it would result in undefined values: $[(x_n)]^{[(y_n)]}=[({x^y}_n)]$ is undefined when any $y_n\notin\mathbb{Z}$
CONSTRUCTION ATTEMPT
Define a "natural exponentiation" function $N$ recursively.
Let $\mathbb{N}$ be the embedding of the naturals in $\mathbb{R}$
$\forall x\in\mathbb{R}\forall y\in\mathbb{N}$
$N(x,0):=1$
$N(x,y+1):=x\times N(x,y)$
This function mimics $^\wedge:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ but the base can be any $x\in\mathbb{R}$
I start the rest of the construction following the post above:
With this we can define the exponential function.
$\text{exp}(x)={\displaystyle\sum_{n=0}^\infty}\frac{N(x,n)}{n!}$
We then define the natural logarithm as the converse of $\text{exp}$
$\text{ln}=\text{exp}^\mathrm{T}$
I didn't prove this yet, but following the post this defines a partial function on $\mathbb{R}$ with the domain being all $x>0$.
$\text{ln}:(0..\infty)\to\mathbb{R}$
And if things were easy, then we could define $^\wedge:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ as follows:
$\forall x,y\in\mathbb{R}(x^y=\text{exp}(y\cdot\text{ln}(x)))$
HOWEVER
$\text{ln}(x)$ is undefined when $x\leq 0$, therefore this definition provided would not work over all $x\in\mathbb{R}$
The alternative answer in the post uses complex numbers. But from a constructive perspective we can't use this approach as $\mathbb{R}\nsubseteq\mathbb{C}$ strictly speaking.
The only way I could think this idea would be helpful is if
- construct exponentiation as above as a partial function over $\mathbb{R}$.
- construct the complex numbers $\mathbb{C}$
- construct complex exponentiation, hoping that it doesn't need exponentiation defined over all of $\mathbb{R}$ first.
- Hope that $\forall x,y\in\mathbb{R}$, the complex exponent of the embeddings $x+0i$ and $y+0i$ is in the embedding of $\mathbb{R}$ in $\mathbb{C}$
- Define "total" real exponentiation of $x,y\in\mathbb{R}$ to be the number $z\in\mathbb{R}$ whose imbedding is the solution to the exponentiation of the embeddings of $x,y\in\mathbb{R}$, $(x+0i)^{(y+oi)}=z+0i\in\mathbb{C}$.
As you can see, that is a lot of effort with lots of "ifs" to construct real exponentiation. If there is another way, or if this way does or doesn't work, please tell.
DON'T FLAG AS DUPLICATE
The post above was flagged as a duplicate of What does $2^x$ really mean when $x$ is not an integer? Neither this post, nor the above address this issue.
