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I'd explain the non standard reals as follows.

The language $L$ contains literally all functions ${\mathbb R}^n\to\mathbb R$ and all subsets of ${\mathbb R}^n$. (Recall: symbols can be anything.)

Consider ${\mathbb R}$ as an $L$-structure. Each symbol in $L$ is interpreted in itself.

Now let $^*{\mathbb R}$ be a proper elementary extension of ${\mathbb R}$.

This makes elementarity and tranfer principle the same thing.

P.S. In the context of NSA (non standard analysis) the real closedness of ${\mathbb R}$ is an infinitesimal detail (excuse the pun). Real closedness is concerned only with the order-algebraic part of ${\mathbb R}$. This is just a tiny fraction of the language you use in NSA.

I'd explain the non standard reals as follows.

The language $L$ contains literally all functions ${\mathbb R}^n\to\mathbb R$ and all subsets of ${\mathbb R}^n$. (Recall: any object can be used as symbol.)

Consider ${\mathbb R}$ as an $L$-structure. Each symbol in $L$ is interpreted in itself.

Now let $^*{\mathbb R}$ be a proper elementary extension of ${\mathbb R}$.

This makes elementarity and tranfer principle the same thing.

P.S. In the context of NSA (non standard analysis) the real closedness of ${\mathbb R}$ is an infinitesimal detail (excuse the pun). Real closedness is concerned only with the order-algebraic part of ${\mathbb R}$. This is just a tiny fraction of the language you use in NSA.

I'd explain the non standard reals as follows.

The language $L$ contains literally all functions ${\mathbb R}^n\to\mathbb R$ and all subsets of ${\mathbb R}^n$. (Recall: symbols can be anything.)

Consider ${\mathbb R}$ as an $L$-structure. Each symbol in $L$ is interpreted in itself.

Now let $^*{\mathbb R}$ be a proper elementarity extension of ${\mathbb R}$.

This makes elementarity and tranfer principle the same thing.

P.S. In the context of NSA (non standard analysis) the real closedness of ${\mathbb R}$ is an infinitesimal detail (excuse the pun). Real closedness is concerned only with the order-algebraic part of ${\mathbb R}$. This is just a tiny fraction of the language you use in NSA.

I'd explain the non standard reals as follows.

The language $L$ contains literally all functions ${\mathbb R}^n\to\mathbb R$ and all subsets of ${\mathbb R}^n$. (Recall: symbols can be anything.)

Consider ${\mathbb R}$ as an $L$-structure. Each symbol in $L$ is interpreted in itself.

Now let $^*{\mathbb R}$ be a proper elementary extension of ${\mathbb R}$.

This makes elementarity and tranfer principle the same thing.

P.S. In the context of NSA (non standard analysis) the real closedness of ${\mathbb R}$ is an infinitesimal detail (excuse the pun). Real closedness is concerned only with the order-algebraic part of ${\mathbb R}$. This is just a tiny fraction of the language you use in NSA.

I'd explain the non standard reals as follows.

The language $L$ contains literally all functions ${\mathbb R}^n\to\mathbb R$ and all subsets of ${\mathbb R}^n$. (Recall: symbols can be anything.)

Consider ${\mathbb R}$ as an $L$-structure. Each symbol in $L$ is interpreted in itself.

Now let $^*{\mathbb R}$ be a proper elementarity extension of ${\mathbb R}$.

This makes elementarity and tranfer principle the same thing.

I'd explain the non standard reals as follows.

The language $L$ contains literally all functions ${\mathbb R}^n\to\mathbb R$ and all subsets of ${\mathbb R}^n$. (Recall: symbols can be anything.)

Consider ${\mathbb R}$ as an $L$-structure. Each symbol in $L$ is interpreted in itself.

Now let $^*{\mathbb R}$ be a proper elementarity extension of ${\mathbb R}$.

This makes elementarity and tranfer principle the same thing.

P.S. In the context of NSA (non standard analysis) the real closedness of ${\mathbb R}$ is an infinitesimal detail (excuse the pun). Real closedness is concerned only with the order-algebraic part of ${\mathbb R}$. This is just a tiny fraction of the language you use in NSA.