The sorcerer's spells known feature says:

You know two 1st-level spells of your choice from the sorcerer spell list.

The Spells Known column of Table: The Sorcerer shows when you learn more sorcerer spells of your choice. Each of these spells must be of a level for which you have spell slots. For instance, when you reach 3rd level in this class, you can learn one new spell of 1st or 2nd level.

The part I'm concerned with is the following: "Each of these spells must be of a level for which you have spell slots." The wording, unfortunately, assumes single-class characters only.

We know from the multiclass rules that if a character is multiclassed between two casting classes, that spell slots are granted based on the following:

Spell Slots: You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, and half your levels (rounded down) in the paladin and ranger classes. Use this total to determine your spell slots by consulting Table: Multiclass Spellcaster.

A level 5 wizard who takes 1 level in Sorcerer has four Level 1 spell slots, three Level 2 slots, and three Level 3 slots, per the multiclass spellcaster table (visible here).

If I'm reading the sorcerer class entry correctly, the first level of sorcerer requires that you pick two Level 1 spells ("You know two 1st-level spells of your choice from the sorcerer spell list."), but every sorcerer level taken after the first can allow the character to learn spells of any level he has slots in, including Level 3 spells such as Fireball.

Is my assumption correct? Can I multiclass into Sorcerer from Wizard and, aside from the initial Level 1 spells known requirement (which can later be swapped with higher level spells per the sorcerer class feature), add spells to my list of Sorcerer Spells Known of any level I have slots in from the multiclass spell slots table?

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    \$\begingroup\$ related: Can a multiclass Wizard use any Wizard spell? \$\endgroup\$ – inthemanual Dec 7 '16 at 18:20
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    \$\begingroup\$ Possibly a duplicate even? The answer seems to be the same: when multiclassing, "you determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class". \$\endgroup\$ – Nox Dec 7 '16 at 18:26
  • \$\begingroup\$ Hmm. The title doesn't seem like it, but reading further the questions do seem to each hinge on matching wording in the two different rules… \$\endgroup\$ – SevenSidedDie Dec 7 '16 at 18:55
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    \$\begingroup\$ Not a duplicate: the Wizard question asks about preparation for a multiclass with two preparation-casters (Cleric and Wizard). This question is asking whether the spell slots gained from a multiclass of learned and prepared levels would allow, a Prepared X who then levels in a Learned Casting class can Learn spells of the slots they have on the multiclass spellcasting table. One (lowest rated) answer on the Wizard question sort-of addresses that, but the other's don't and the Wizard question didn't ask for it. \$\endgroup\$ – Rob Stening May 2 '17 at 17:11
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    \$\begingroup\$ Ah, apologies. I overlooked that. The question itself still isn't duplicate, but does happen to be answered in an answer. Okay. \$\endgroup\$ – Rob Stening May 4 '17 at 16:45